Here are some networking tips from screenwriter and story development guy, Neo Edmund:
He began with no Hollywood connections, but attended a pitch fest in 2001, then interned at the pitch fest the following year.
Networking is not about "read my screenplay," it's about "friending." People who hang out and make friends usually do better. Attend a networking event a couple of times and people will usually ask you where you come from.
He spends time networking through Xbox, which involves chatting online and playing video games.
Nobody exchanges business cards anymore, it's all about Facebook.
He holds a once-a-month by-invitation-only networking event. A guy came with a knapsack full of scripts---he took him aside and told him that's not really the way it works.
How do you contact someone you connected with at an event but with whom you didn't exchange contact information? One word, "Facebook."
Follow up the next day after the initial meeting with a note saying, "Nice to meet you, this is what we chatted about, and hope to see you again."
If you do finally want to ask a contact to read a script, after a friendship connection has been made, Edmund suggests having a coffee date. Be easygoing and ask if the other would be willing to give you feedback. Oftentimes your contact will be glad to help.