Working together (even if separately) with other people is highly motivating. I found that I was much less likely to procrastinate this way, kind of how I'm less likely to be reading Hacker News instead of working when my boss is standing behind me than when I'm alone.
Personally I don't think you need a whole separate website for this; a Discord or IRC bot that runs the pomodoro would work just as well.
When someone else is around it's like a switch gets turned on and BAM - I'm able to do the thing.
Maybe a purely structured chat would work - summarise what you achieved, outline what you will do in your next allotment. But anything beyond that might be time better spent planning and prepping for the next session rather than wasting the first minute getting up to speed?