There are female-only Grand Lodges, and mixed sex Grand Lodges. I regard lodges that accept women as legitimate (but not regular) if they are recognized by a reasonable body of female masons or co-masons, assuming they keep the other Ancient Landmarks, but I will not sit in Lodge with them nor allow them to sit in my lodge. I'd work with them on charitable events and for mutual protection, and even enjoy socializing with them. I would respond to the Grand Hailing Sign given by a woman.
Continental Freemasonry (or Oriental Freemasonry) follows the Grand Orient of France, sort of like the Grand Lodge of France, only different. French Freemasonry split over whether to recognize atheists as masons, the Grand Orient accepting atheists, and the Grand Lodge refusing. These Freemasons may be mixed-sex or not, but they do not require belief in a Supreme Being. They are popular in Europe and Latin America, and in Lebanon. British-based Freemasonry (like us) do not recognize them as masons.I regard Continental Freemasonry, if in amity with the Grand Orient of France, as legitimate (but not regular) assuming they keep the other Ancient Landmarks, but I will not sit in Lodge with them nor allow them to sit in my lodge. I'd work with them on charitable events and for mutual protection, and even enjoy socializing with them. I would respond to the Grand Hailing Sign given by an Continental mason.
I need communion with other men, framed in ritual, and devoted to Brotherly Love, Relief and Truth. I go to a non-Orthodox synagogue twice a week, and pray with women and there are female officers and a female rabbi and I love my religious community. I get mixed-sex communion at my place of worship. For Freemasonry to work for me, and to give me what I need from it, I need to sit in a lodge of men. Women get the rest of my life, but not my lodge time. I think that men need this structure to help them become better men. Better people as well, but particularly better men.
Men all over the world are starving for good role models of sane masculinity. Boys put themselves in terrible danger trying to prove themselves to their peers with acts of recklessness and false bravery. Our society lacks good initiation rituals, and without initiation, a man does not know he is a man. A woman's body initiates her throughout her journey. She grows breasts and hips and knows she's not a child anymore. She becomes menopausal and knows she's not a young woman anymore. Men lack this, and ritual takes the place of biology for men.
High school graduation does not initiate manhood. A bar mitzvah is not sufficient. Eagle Scout ceremonies miss the mark (although the Order of the Arrow is pretty good at this). A youth needs to walk into a room, or into a forest, or into a cave, and walk out a man, through the help of older men not related to him who have gone through the same ritual. When men are not given a proper initiation, they turn to playing chicken with cars, gangs, petty crime, drugs and alcohol, predatory sexual behavior, other ways to prove themselves. Freemasonry is the best structured method I know to initiate men into the right kind of manhood.
I have many friends who are atheists or agnostics, some of whom would benefit from Freemasonry, but would never petition because of their inability to submit to a Supreme Being they cannot conceive of. I'd rather have such a person benefit from Freemasonry to the best of their ability than stay in the dark. My own prejudices lead me to regard Continental Freemasonry as inferior to Ancient Freemasonry, but I'd rather these Brothers and Sisters enjoy partial light than no light.
Update: I feel it is important for me to make three disclaimers here. The first is that I will not consider sitting in a tyled lodge with any but regular masons, whose Grand Lodges are recognized by the Grand Lodge of Massachusetts.
The second is that I have nothing to do with rogue groups like the Grand Orient of the USA, and the several hundred fake Prince Hall Grand Lodges (although I'd like to see some sort of resolution between PHA and PHO). Masonic diploma mills are beneath contempt, as are fake appendant bodies.
The third is that it was pointed out to me that the Grand Orient of France and masonic groups affiiliated with them are actively involved in politics, have been involved, as masons, in revolutions in Latin America, and do things like protest the Pope's visit to their country. I regard such behavior as fundamentally unmasonic, and utterly toxic to the fraternity. Such behavior gives anti-masons fuel for their hatred, and destroys harmony within the lodge, and between lodges.
The argument is made in many places that atheists cannot swear oaths because without belief in Deity there is nothing backing the oath. Thus an atheist cannot make masonic Obligations. I do not subscribe to that argument, although I respect the beliefs of those who do. I can understand, and I totally support those who feel that without belief in Deity, Freemasonry is impossible. I do not regard an atheist as a regular mason, and woud never consider joining a masonic lodge that admitted atheists, but not because I feel they are incapable of Obligation. My reason instead is because the blue lodge is dedicated to God. It is the Temple of Solomon rebuilt. Masonic light is given us by the Great Architect of the Universe, and by His light we see light. Without Deity there is no altar, and therefore no lodge. The circle ceases to have a point in its center.
That being said, I know people who call themselves atheist or agnostic and are very spiritual people. They, in general, seem fixated on Deity being an angry, bearded anthropomorph in the sky, who sits on a throne and is mad at everyone for following their lower chakra urges, and they utterly reject such a grotesque caricature (as do I). I am convinced that if some of them could enter my mind and see what my conception of Deity is, they would slap their foreheads and admit that such a Presence was on the threshold of their experiences all their lives. I would want such a person to have experiences that would lead them to understand that the focus of their spiritual intent was the very thing they rejected, the stone that the builder refused. I won't open my West Gate to them, but I'd love for their experiences to lead them to some sort of West Gate, when they are ready.
I would never want to see the militant atheism, or antitheism, of Richard Dawkins, Christopher Hitchens, or Sam Harris fall under the banner of Freemasonry. I would fight that with everything I had, within the bounds of the law, of course.
Internally, I've been making an analogy to Judaism, in which there are three major camps, and several minor ones. Orthodox Judaism applies Rabbinic Law absolutely, Conservative Judaism tempers it for modern times, and Reform Judaism chucks out the law and replaces it with looser creeds. Minor camps include Reconstructionist Judaism, which splintered off from Conservatives in favor of a panentheist view of God, and a sense of Judaism as a civilization rather than a set of codes of law. Jewish Renewal, my denomination, seeks to focus more on spirituality than law, but allows individuals to follow the law at their own level, subject to their individual consciences, but wants to be as inclusive as possible without destroying what makes us Jewish. Humanistic Judaism rejects Deity completely, and sees itself as a cultural organization. Messianic Judaism accepts Jesus Christ as the Messiah (and therefore technically isn't Judaism). Modern Orthodox Judaism seeks to remain Orthodox while still embracing modern culture that more strict Orthodox Jews avoid. Conservadox Judaism is another group most of which came from Conservative Judaism and wanted something stricter, while a minority came from Modern Orthodoxy and wanted something looser.
I regard myself as a Conservadox mason. What separates me from an Orthodox mason is that I don't feel that Anderson's Constitutions, or the Preston-Webb monitor is the final word in Freemasonry, the same way that Orthodox Jews feel that the Torah, interpreted through the Talmud, is the final word of God. I imagine that the fraternity will go through significant changes in the coming decades, and I don't entirely dread this if it is done properly. I would happily sit in a tyled lodge with Conservative or Orthodox masons, assuming their Grand Lodge was in amity with mine. If I ever sit in the East, the first time I would run the lodge as Orthodoxly as I could without upsetting the Brethren, and afterwards I would follow my conscience within the bounds of my Obligation. For example, I would not turn a visiting brother away for lack of suitable attire.