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.
I have multiple projects that use variations of the same base workflow. The variations depend on the requirements of the project or issue type. The variations mostly come in the form of new statuses ...
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