Is difficulty swallowing from a tracheostomy permanent?

Yes: If the tracheotomy is permanent, then yes, the swallowing problem is permanent.

Soemtimes: The problem with the trach is it ties the voicebox down so it can't do the normal movement it does with swallowing. Lots of patients who have a trach but aren't on a breathing machine eat through their mouths once they learn how with the help of swallowing therapy. There are also appliances that can help with learning how to do it anew.

Usually not.: When you swallow, the voice box bobs up and down (watch someone's adam's apple when they swallow). The trach tethers that "bob", but over time people get used to it and swallowing returns to normal.