I first went there when it was called "Pico-Burnside Baths" ('cause at Pico Blvd at Burnside) - sort of like an old Russian baths. Ones "old school" straight men's baths, like in a basement in New York, Boston, or Chicago.
Name change to CITY SPA ..... and lots of $$$ put into it. Wet area was all men (nude): steam, 2 saunas, showers, whirpool, and swimming pool (cold plunge added with CITY SPA remodel) .... tanning deck. With a clothed gym area (small) and cafe. Then it began to have "co-ed" days, swimwear required on those days.
So it has no seperate facilties as per your question. I believe their web site gave which times were co-ed and which days were men.
On some other message boards there has been a buzz about CENTURY DAY SPA:
http://www.centurydayspa.com
Seperate men's and women's facilities.
They have ads in LA WEEKLY and also METRO SOURCE LA (the gay publication).
It a Korean Baths at 4120 W. Olymic Blvd., LA