Open a contact and click the button (towards the top) that says "Request Update".
You'll then get a form (you can edit the template for this one under the email templates), but you can change things here or use what's already there. Add in your signature and click Send.
What they see on their side is an email that lists out their information and has two links where they can update changes or leave as is - either way you've got confirmation. The link goes to a live Salesforce page where they can update the information. They can also drop you a note at the bottom of that page (kind of nice).
Once they do this, you'll get an email to review the page then update the information in Salesforce. The email will let you know that he/she verified it (if no changes) or it will have changes, with a link at the bottom of the email to go ahead and update the record.