I have been using small groups in Zoom, typically 3-4 students per group. Typically random groups, but also more open and longer term tasks. Sometimes with a few minutes of individual work first, sometimes without. This has worked out fine, with people even using cameras a lot of the time when they are in those small groups.
It is quite a lot of burden to ask them to figure out their own means of establishing a workspace. You should teach them how to do this or you should make it as automatic as possible (which Zoom makes possible). The more they have to think and choose and mess around, the easier it is just to not bother and instead work by oneself.
Also, if the exercises are very short (a few minutes), the time it takes to set up an external communication is long when compared to the length of the task. Lots of friction there. And with only two people, they both have to be awake and active enough before communication can happen.
My concrete suggestions:
- Use Zoom to set up the groups.
- Have groups of three to five people, so that even if some of them are not active, the others can still collaborate.
- Give clear but open-ended tasks or tasks that allow several approaches. This helps in creating mathematical dialogue. If the tasks are simple calculations there is a big risk that someone solves and shows, or that several people work in parallel without much discussion.
- Have tasks that are long enough for the students to figure out that they should be doing group work, figure out what they should be doing, get started doing it, and have some time to do it, too.
- If you use Zoom functionality, you can cycle through the groups, seeing what people are doing there, whether they understood the activity or not, how far they have come, and there is also the opportunity to have a discussion with them. I usually start with the smallest group, the group I guess might need the most help, or a random group, and visit them briefly starting from there. After a few groups are finished or have accomplished the main points (always good to have some extra activity for the fastest groups) I typically end the small group sessions. Then I select a random person or a random group and ask them to report what they have accomplished, asking if others have comments or commenting myself. This keeps the students accountable.