When I was a teenager, I went to a hunting-themed restaurant. When it came time to visit the restroom, I saw that the signs on the doors depicted bird dogs. One door had the picture of a pointer and the other the picture of a setter. I chose the door with the pointer on it, and when I got inside, I found a line of urinals along one wall. The restaurant's humorous reference to the structural differences between men and women made it very easy to distinguish the men's room from the women's room.
Oddly enough, restrooms have historically been designed to accommodate the external plumbing of the "gender" using them. Lately, in a move for gender equity, buildings have begun installing "family" restrooms which are essentially unisex rooms designed to accommodate parents with small children. Now the ruling has come down from Mount Olympus, er Washington, that the external plumbing of the users of bathrooms is irrelevant to whichever bathroom they choose to use, and if the recipients of government grants want to keep on getting government funds they'd better get with the program and let people use whichever bathroom makes them feel better.
Transgendered people have been around at least since the seventies. When I was a public defender, I represented a woman who disclosed to me that I had gone to high school with him. It was something of a shock back in that day and age, but my client had submitted to the necessary medical procedures and was living with her boyfriend. A few years later an acquaintance of mine moved away from town, had a sex-change operation, and became a lesbian.
I never followed either one of these people into a bathroom, but I don't think they caused a stir by entering a women's room and using the facilities as "setters." I don't really see how something like that could be aproblem unless the transgendered individual marched into the bathroom beating a drum and loudly proclaiming "I am transgendered!" That seems to me to be about the only way anyone would notice, unless the transgendered individual was some sort of a reality TV star.
The very real problem with letting people equipped with external plumbing who claim that they "feel like a woman" use the women's room is that there are a lot of dirtbags among the ranks of the pointers who would be happy to use such a license to go into women's rooms for the purpose of voyeurism and sexual harassment.
I propose a gender-neutral solution to the problem. Take down all signs everywhere which refer to the "gender" of the users of the bathrooms. No "Men's Rooms," no "Women's Rooms." Put up signs saying "Pointers" and "Setters." Those who came into this world fully equipped for pointing and who have not had that equipment surgically removed, regardless of "gender," would be required to use the "Pointers" room. All others could use the "Setters" room. Those who feel uncomfortable in either place can opt for the "family" bathroom.