Technically anyone who's worked for SHIELD has at one point been a puppet to HYDRA. Hawkeye was of their top agents even before Widow joined and, in retrospect, had to carry out their dirty work for them.
I was actually going to bring up that it might have a particularly cruel punch to it for Black Widow as well, especially considering her line in TWS about trading in the KGB for SHIELD and not really knowing who lies she was telling. On the other hand, Hawkeye's situation is obviously a little different as not only did he have to deal with SHEILD's collapse and learning of the HYDRA infestation but he was literally brainwashed into playing Loki's puppet. So while I think Natasha's character would feel the personal sting of those lines, I think he focus is likely to be on Hawkeye.
I mean assuming these lines are actually true to the film and not simply being used as part of making an awesome trailer, it fits very well thematially to put focus on Hawkeye's reaction to being called a puppet precisely because his story has sort of been on hold since the last Avengers film, and while the characters will have all been changed by their experiences of the past films:
A. Hakweye doesn't have that filler, so audiences are going to associate him with the last time they saw him.
B. In A1 as I'm sure will be the case with A2 Joss did a wonderful job of utilizing the characters as if they had gone through all their various solos while also limiting his reliance on the audience needing to have seen any of those films. So while I imagine Black Widow's personality (and of course the setting as a whole) will have been effected by TWS I don't expect that Whedon would heavily rely on precise scenes/character arcs. In other words, he might show how that line gets to Natasha without lingering on specifics. Hawkeye's a different story as his big "puppet" moment was in A1 which I think Joss will be more open to grabbing from.
C. Because I think the first movie will be more specifically referenced than the solo films
and because Hawkeye has no real character movement in that film
and since one of the themes (at least as presented in the trailer) is free will verses being controlled, it would be a great way to utilize what happened to Hawkeye in the first movie to create character drama. In a way it almost retroactively helps what was admittedly poor usage in A1 if that experience were to somehow directly effect his character and reaction in A2.
If you watch A1 now it's just like, "Oh, Hawkeye is a zombie through most of this" but if after A2 you watch it and think, "Oh, that explains the way he takes Ultron's comment so personal/the way he is able to talk to the twins when they are being controlled by Ultron/Why he struggles so hard and reacts like he does to SW's 'playing' with his mind" it won't fix Hawkeye's treatment in A1, but it would sort of help make it more easy to justify if in A2 it really helps to push his character development. At least then his time as a puppet isn't merely for convince of the plot, it actually means something to the character and the direction they go in.
Which isn't to say it will happen and it is something that is definitely and utterly true, just that I honestly think it could be a good way to move the character and the way they cut the party scene in the trailer really does appear to me to emphasis on Hawkeye's reaction during the scene. Not just the way he seems to be on guard and ready for action it also happens to use a clip that's reminisce of a brainwashed Hawkeye from A1 after pointedly showing us a very different man earlier. I just think it has the potential to be an interesting character moment that could just as easily be nothing with his time in A1 never mentioned.