I didn't do mere BBQs but threw full-on parties with the neighbors (who I didn't know at all) and other friends. Later two shared apartments in the same house combined held a huge party that spanned the house and included many of the neighbors. Many good friendships came out of that, and a couple of us moved in together later.

The BBQ idea is just a low-threshold variant of that which doesn't require copious amounts of alcohol.

For free stuff, we just have a place in the staircase where people drop things that are still good but not needed by their previous owner (mostly books). This works with zero explicit coordination.