Is there any
OOTB way of seeing items in the Recycle Bin deleted by other users? Or do I need to wait until they appear in the
Site Collection Recycle Bin?
As a site collection administrator, go to the site collection recycle bin, from there you can see either all the items that users have deleted in the site, it's subsites and all the lists and libraries, or see items that the users deleted from their recycle bin.
As a site collection admin, you can have access to both levels of Recycle Bin. If user delete a Item/List or library then it will go to End User Recycle Bin and you can see that there.
But User Delete the Subsite then it will go the 2nd Stage recycle bin and you can see it there.
Aso note one thing, Recycle bins are time sensitive and quota limited.
Another Thing, if you enable the auditing on your site collection, then you can see who did what.
A SharePoint site collection administrator can view and restore deleted items from the site collection Recycle Bin to their original locations. If an item is deleted from the site collection Recycle Bin, or it exceeds the retention time, it is permanently deleted.
If you are a site collection administrator you can see the two levels of recycle bin... in "End user recycle bin" section, you can see the items deleted by another users.