Song sessions and tune sessions can work very well without any kind of formal structure, and that can go for the case where you have the odd song in a predominently tune session.

But I think if you're having a mixed session where the tunes and the songs have an equal place, it's probably best to have someone in the chair saying "You next?", with an inbuilt sense of balancing the evening, and bringing in the newcomers and so forth.

As for the problem of people joining in when you'd rather they didn't (or the other way round), I think it's a good idea not to rely too much on telepathy. Say what you want, and that is probably what you'll get.