Allow access to upload files without “edit others” capability granted

I want my users to be able to upload multiple images in their front end posts. I’m using a plugin to do this and enabled the visual editor with “Add Media” button. However, in testing the different combinations of capabilities, I’ve found out that in order for a user to be able to upload their pictures in this way, you have to allow them the ability to edit other users’ posts. I definitely do not want this. Is there any way I can allow users to upload photos in the visual editor without giving them access to edit anything?