I don't think Asana does this in a "simple" way, it's a complete mess from what little I've seen of it.
The answer for JIRA depends on how you're using it. JIRA Core is business oriented, with a lot of ways of seeing what users and teams are doing. Software adds a load more. Service Desk specifically implements queues for teams. None of them do a "simple" user-is-doing-x because it's not really any use in isolation.
We need to know what your processes are, and how you're representing them in JIRA.
Asana is still a mess because it can only answer one relatively useless question at a time.
Start with a search for the issues you are interested in, and include assignee in the results. Have a look at the developers boards, look at the project reports, create a dashboard that shows you all the data overall.
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