It is good to let them know that you know lots of people that they know. But let them talk directly with your references.

Think of this as a basic social situation. If I tell you, "Fred thinks that I am a great guy, and here is what he says about me..." what do you think? If I say, "If you want to know about me, ask Fred." and then you ask Fred and hear what a great guy I am, what do you think? If Fred was talking in front of me, and I walk away, do you feel tempted to ask Fred, "OK, now that Ben's not here, what do you really think?"

When you hear compliments about me directly from Fred, it carries a ton more weight, doesn't it? If I'm repeating lots of good stuff about me directly to you, it is natural for you to wonder if I am accurately repeating everything that Fred really would say. Furthermore you are going to trust what Fred says about me more if I am not present.

References work the same way. People will most trust a reference which you didn't present, which they know that you didn't see.

Cheers,
Ben