Trust is a result of authenticity. In order to leverage your network to attain introduction to prospective employers, you must create trust.
Let's assume you have set up a coffee or lunch date with someone in your network who has a connection that you want. This conversation needs to be much more than an exchange.
You want to turn this person into your advocate. At a minimum expect to have 3-4 interactions with him or her over the next two months, possibly more. Don't expect any intros immediately during or after the first interaction.
1. Appreciation/Acknowledge the situation - The start could be a super awkward if not done right - you can control that! So take control of the meeting. Appreciate his or her time, appreciate their willingness to support you. Acknowledge that he or she is taking the time from her day to help you.
2. Context/History - very clearly and concisely explain where you are at in your career.
"This is what I love about my job _________, I want to do more of that in another company"
"I add the most value by _______, I want to do more of that in another company"
On the call focus on the things about you that position you to add value to a company.
3. Questions - Does this person in your network have specific experiences you want to ask about? These should be real questions (not questions just for questions sake, if you don't have questions you should wait to have the call until you do)
4. Reciprocate - Specifically ask how you can help him or her now or in the future. What are his/her needs?
5. Action - Always end it with appreciation and some sort of next action step. In this case I would bypass my usually recommendation of "Is there anyone in your personal or professional network you think I would benefit from getting to know" because that is the focus of your conversation. You could end it with, "Can I take some time to research some of the areas we talked about - and follow up with you mid-next week, most likely asking for an introduction"