Assign an Issue to make it clear who's looking after it.
On the Active tab, find the Issue.
Click the + at the right of its row.
Choose a team member, or Unassigned to clear it.
The row then shows that person's profile picture.
Assigning an Issue doesn't notify the teammate. Let them know yourself, for example in Slack.
To add someone to your team, see Invite a team member.