Depends on the festivals of course, and what they're offering. Some actually have schedueld 'meet and greets' - where they line up producer with writers. Some are more ad hoc. I've made connections by going to a lecture by one person. Then a DIFFERENT person in the audience introduces themselves to ask a question. Afterwards, I go up to the person from the audience, and ask them if they have a few minutes, or if I can buy them a coffee - I'd like to talk to them about a script. Sometimes we chat, sometimes they just say "Have you got it with you?" (The answer should be YES. I always attend events and meeting with at least three hard copies, and a thumbdrive copy)
It's about polishing your social skills really. Be polite, be to the point, and be ready to shake hands and leave them with only your card. A good impression is important.