Quick answer: In Gateshead you can play board games at Board game sessions at ToastieLabs (games event) and Board Game Sundays at Baltic (games event, Sundays, 10am to 5:30pm). The next confirmed date is Sat 14 Nov, 10:30 am (Board game sessions at ToastieLabs). One more group is listed below; check their own page before you go.
Baltic on Gateshead Quays puts out free board games for all ages every Sunday, and ToastieLabs on Half Moon Lane runs board game tasters and family mornings at its gaming tables. Gecko Gaming in Felling also has a board game library. Newcastle, across the river, has more going on.
ToastieLabs is an indie games studio with a hobby shop and bookable gaming tables on Half Moon Lane. It runs board game sessions, including free try-before-you-buy tasters and family board game mornings.
A games shop in Felling, mostly known for wargaming, with a board game library that's included in the £5 table hire and a board game night on the last Tuesday of the month.
Where
97-99 High Street, Felling, Gateshead, NE10 9LU
When
Saturday 11am to 7pm, Sunday 11am to 5pm, plus gaming evenings in the week
Cost
Table hire £5 per person, no time limit, board games included
Games to borrow
Yes
Check the shop's own page before you go, as times can change.