Hello @christina.owens and welcome to the Asana Forums!
One of the ways my organization accomplishes this is using the Status Update feature on our Asana projects and then grouping projects into Portfolios. Here is a quick video that outlines how to use this function and here is a guide with step-by-step instructions on how to do this.
The key for this is that you will need to instruct your users to provide status updates for the projects that they own. You can do this on a project-by-project basis or you can also create a project for each team member (you could name it something like Bill’s Task List or Monica’s Status Updates) and have them update the Status on that project. From there you can quickly and easily see all of the status updates across all the projects in one place grouping all of the projects into a Portfolio and then sharing that Portfolio with everyone on the team who you want it to be visible to. Here is a guide on how to set up a Portfolio.
Hopefully this helps but if anyone else on the forum has another way of accomplishing this please share!