Is it possible to assign individual permissions through some method to keep site members from using each other's images.

Same here.
It's would be nise feature.

I agree. I like the module, but I can't allow all users to have access, because it could be used to break the site. But as an admin, I'd like to have it for my own use.

IMCE profiles are assigned to roles, and you can create specific roles for specific users. You can allow profile P to browse but not to upload or delete in folders assigned to profile Q.

Edit:
If you are asking for a file access system you should be using private file method and implement your own file access rules.

Thank You for help.

The majority of the site I need this ability for needs public files -- we use IMCE for inserting images, PDFs, other files across the entire site and it works well. Hundreds of nodes have these kinds of files inserted into them via IMCE and changing the file system to private would break these links.

However, we do want to allow users to upload to their own directory (which they can already do thanks to IMCE Profiles) and users can not browse each other folders or anything like that -- but I still don't want anything inside users/[uid]/ to be accessible via http without someone being logged in with the approprate "admin" role. Does anyone have a suggestion for this?

I am creating a marketplace website. Each seller should be able to upload their own images and other users should not be able to access (in file directory) images uploaded by others. How can I achieve this?

As of now, when logged in as admin (user 1) the file directory shows the root directory.