Absolutely they could be sued, since it's very much illegal (whether you're using it commercially or not). In practice, I think it's unlikely. Publishers have bigger fish to fry. I could see them suing cam sites as opposed to individual models, but in that case, if the site complies with Safe Harbor, that might not get them far (though it's muddy considering they pay us, I'm not sure I've seen that scenario come up before).
As mentioned above, some sites do play it safe and disallow music. I think I've heard of one or two that even require brand labels on clothing or food items etc, to be covered up, which I find absurd. Streamate doesn't allow music during TopSpot shows, I'm assuming because they repost them elsewhere themselves, and so in that case they would be directly liable (I'm not entirely sure what is done with those shows anymore, though).
Personally I would never use music I didn't have rights to in any of my videos, but I don't worry about it when I'm streaming live between shows. I typically turn my music down or off when in a private show. Not even for copyright reasons, I just think it's distracting for the viewer. I'm usually listening to something annoying... My music is usually more for my benefit. It's uncomfortable for me to talk out loud and have all the responses be silent and typed. Music makes me feel less like I'm talking to myself.