I imagine some homeless people do, but not many.

It's a system largely based on reputation and feedback. Being unwashed, having a lot of stuff, not ever hosting, or having negative feedback will throw up red flags for hosts.

(Not saying that homeless people are necessarily like my description, but I imagine those would be factors in being rejected).

I used it while homeless and woke up to an unwanted sexual experience and never used it again.

I've hosted one homeless person. She was living in her car and it was a temporary thing. In big cities, getting requests from car dwellers who need a break is fairly common. Some people list in their profile that they only host travelers.

It's not exactly a service. Of the people that would request to stay with me, I would probably say yes about 10% of the time.

If it was convenient, if they seemed interesting, respectful, etc.

Great question. Homelessness is similar to being broke and in a new place traveling as a tourist.