Find user mention through:
Assigned checklist items page
To see items where you or somebody else is mentioned navigate to Assigned checklist items page.
You can find the link to the Assigned checklist items page in your personal menu (click avatar in Jira's sidebar or top navigation bar):
The page displays a list of items (grouped by issue) in which the selected user is mentioned. Items are in read-only mode.
The checklist rendered under an issue contains only those items in which the user is mentioned. To see all items in the selected issue, navigate to the issue by clicking on its key or summary.
The items in which the user is mentioned will be displayed even if you don't have permission to view checklist in a given issue or permission to view the issue itself. In the latter case, you will not see the summary of the issue, just its ID.
User picker and shareable URL
Only Jira administrators can use user picker and see items assigned to somebody else.
If you are not a Jira administrator user picker will be disabled, otherwise, you will be able to select a user to see items assigned to him/her:
After a new user is selected the URL will be updated to contain the information about selected user. You can then share this link with other Jira administrators. If a user who is not an administrator tries to use such link to see items assigned to somebody else then him/her the user will see a warning and will be offered an option to see items assigned to him / her instead:
Search advanced with JQL documentation page shows an example of how to find Jira issues with a checklist mentioning selected user.