Does anyone know how to separate these downloaded mockups so I can work on them individually?

If that object is “Hoodies.png” then it is a single PNG image file (pixels), not a vector file containing multiple objects.

So if you want to work with the two sides separately in this Illustrator file (like, if the source has no vector version available), you will need to split that image in two. Your options are:

Duplicate Hoodies.png in Illustrator, and hide the half you don’t want in each image so that one shows only the left hoodie and the duplicate shows only the right hoodie. You can hide parts of an image using Object > Crop Image, or by using a drawn rectangle as a clipping mask.

or

Edit Hoodies.png in Photoshop, cutting it into two images so that each contains one side of the hoodie, then bring those two images into Illustrator separately.