Yes, for Jira Server the Intercom app is part of the Jira integration. Have a look at the getting started guide on how to set things up with 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 and comment Jira issues. Additionally, if you are using Intercom for Jira there are also features to link and unlink Jira issues to and from Intercom conversations. Currently, Jira Cloud for Intercom only connects to Jira Cloud instances.
Intercom for Jira Intercom for Jira is a paid Jira app that allows users to link Intercom conversations to Jira issues (directly in Jira or from Intercom using Jira Cloud for Intercom). Intercom for Jira can post issue updates back to linked conversations, so Intercom users keep up to date. If an issue gets resolved, linked conversations can be reopened automatically, so you 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 of those applications there is no way for Jira Cloud for Intercom 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. If you would like to change the Jira issue reporter for issues created via the app follow these steps on how to change the issue reporter.
Jira Cloud for Intercom is completely free of charge. However, note that features like link Jira issue and unlink Jira issue are not very useful without the Intercom for Jira integration (paid Jira app that facilitates a workflow between Intercom and Jira).