Following are some of the cafes in Pune where I went and played chess with friends. So most of them will allow if group is small (2-4 people)
If your group is big or if you are planning a meetup, it’s always advisable to ask first.
1- Third Wave Coffee, Kothrud (Near Karishma)
2- Third Wave Coffee, Phoenix Marketcity, Wakad
3- La Casatta Cafe, Near deccan gymkhana
4- Starbucks Kothrud