No they don't, they have an open wound.

Why, this is fascinating! So they can piss, shit and lay eggs out of the same hole? And to think all these years I’ve kept chickens, when I could have had trannies!

This is offensive to birds, who are arguably far more human than trannies.

A quick search on duckduckgo didn't disappoint: The word cloaca is from the Latin verb cluo, "to cleanse". Therefore the noun means "sewer".