I used to have a course mailing list that subscribed every student in the class. Students were permitted to ask and answer any question on the list at any time. I encouraged people to help one another using this list.
The list was used quite a lot by people when they were doing their homework, and they wanted quick answers. I would try not to answer immediately so that students would have a chance to chime in. With such a list, you can even respond to a question by asking for answers from anybody monitoring the list.
I learned over time that about half of the answers came from me and also that I would have to occasionally issue a correction to an incorrect answer. I might also comment on the excellence of a post if appropriate.
Basically, it just worked. I didn't have to offer points and such to encourage participation.
However, you will find that most of those giving answers are the same people. I think it is harder to get some people to participate. Participation was far from uniform.
One advantage of such a list is that even if you, the student, didn't ask a question yourself, you get to read it and one or more answers. And another advantage is that students don't get stuck between class meetings and stop progress when they could have been learning. People could work in the middle of he night and still get questions answered in a timely way.
This has a different dynamic from asking for solutions in a face to face situation. However, I think you will find that it is the same people willing to answer, rather than a uniform spread of participants.
If you want something that will encourage wider participation during class time, make sure that every student has a few index cards at all times. When a question is asked, ask the class as a whole to write a hint for the answer on an index card. Collect and scan these, picking out one of the good answers to read. You can make it anonymous or not. But note that it takes some time to do this so it works best with small groups and flipped classrooms where participation is the main goal of the face to face sections.
I've found in general that index cards have so many uses in teaching that I basically required their use; note taking, for example.