Yes, for Jira Server the Intercom app is part of the Jira integration. Have a look at the Get started guide for Jira Server.
Jira Cloud for Intercom This is a free-of-charge Intercom app that helps Intercom users interact with Jira issues. It allows them to create new, and link existing Jira issues. Additional features such as commenting and unliking Jira issues from conversations are available if Intercom for Jira (see below) has been installed.
Intercom for Jira Intercom for Jira is a paid Jira app that allows users to link Intercom conversations to Jira issues and post issue updates back to linked conversations, so Intercom users keep up to date. If an issue gets resolved the app can reopen linked conversations automatically, so Intercom admins know when it's time to get back to the customer.
You can fine-tune and control access to Jira projects (and other Jira resources) by setting up a dedicated Jira user for Jira Cloud for Intercom. After that, you can configure permission for this users in Jira and this way control what's exposed in Intercom. Have a look at the guide on how to restrict the app's access to specific Jira resources.
Intercom and Jira are two independent applications. While your employees may have a user account in both applications there is no way for the Intercom app to know which Intercom user matches to which Jira user. The issue reporter will always be the user who was logged in to Jira when you first set up the Intercom app (we call this user the app user). If you would like to change the Jira issue reporter for issues created via the app follow these steps on how to change the app user.
Jira Cloud for Intercom is free of charge. However, note that some features are only available if the Intercom for Jira integration has been installed (paid Jira app that facilitates a workflow between Intercom and Jira - see also: What's the difference between Jira Cloud for Intercom and Intercom for Jira?).