Couchsurfing
PaidStay free in locals' homes, live the culture from inside
A cultural exchange network more than accommodation; hosts expect conversation and shared experience, not money. There is a small membership fee. Reading references carefully and filling in your profile are the basis of safety; the Hangouts feature also enables spontaneous traveler meetups.
Stay with a local host in Tbilisi; open khinkali with the family instead of a tourist restaurant.