Correct.
A deal to favor your trading Partner exhibit A:
March 2009: NE gifts the new KC GM Scott Pioli a present:
QB Matt Cassel, and one if it's All-Time Clutch performers, LB Mike Vrabel, for a paltry 2nd RD pick.
I understand your premise, but this is probably a poor example.
At the time, after the Pats had tagged Cassell, they were so tight to the cap that they had no space left to sign their own in-house UFAs until they got Cassell's number off the books. They were behind the 8-ball an all their potential trading partners knew it. They weren't doing Pioli any favors, probably more the opposite.
In the case of JG, my guess is that they
may have gotten
somewhat better offers from a couple of AFC teams but simply didn't want him going to any other team in the conference and they took the best deal they were offered by an NFC team. That may have come down to SFO and ARZ, since most of the other NFC teams appear to have a a significant investment in a "starter" for the next couple seasons. If the SFO-ARZ offers were equal, SFO may have gotten the nod simply because they have more cap space to proved JG with an appropriate extension ($57 million vs. $5 million for ARZ).
That said, I think your premise may be somewhat applicable to lower level deals on occasion, and it certainly seems to apply to mid and late round picks-for-picks trades during the draft where the Pats have often given a bit of a discount off the standard value chart "price" to their regular trading partners.
In any case, I really don't think your premise applies very well to trades involving premier player and top-40 draft picks like the JG deal.