That's right. The other team doesn't get the TPE from the Jazz in the trade, although the trade may generate a different TPE for them as well. Example: Jazz use part of Harpring $6.5M TPE to acquire a $3M player from Team B. Team B gets a new $3M TPE in the deal. Jazz keep the remaining $3.5M of their own TPE.
What's this about another team signing Matthews and trading him back to Jazz? That can't work. Teams can only S&T their own free agents.