The only permission needed is translation management rights on one of the target languages - they do not need any permissions on the items in the job. The reasons it mentions you need "one other right" is because TM is an extension of CM. And if you do not have a single CM right then CM will hide the publication from you as it does not see the TM right.
One thing that is somewhat confusing is that Translation Management rights are always resolved according to the target publication. It's fairly clear if you see it as "I need rights to be able to modify an item" as a translation modifies the target and has no impact on the source. But if you think of it as "I need rights to do something to the item I have selected" then it won't do what you expect.
There is a configuration value in TranslationManager.xml where you can configure which user TM uses for the actual translation work. Default the installer will use a system admin. In larger systems I would consider creating a specific Translation Manager user - that makes it a bit clearer in the item history, and it allows locking down rights a bit tighter if needed. If changing the user, remember to undo checkout of items in translation for that user. If you don't do this, TM will fail on retrieve as it won't be able to change an item checked out to someone else.