UPDATED: S02E04 has some throw-away dialog about this topic; I've updated my answer as spoiler-free as I could.
There are two different "seeing Wesen" effects going on here; one for Grimms and one for everyone else.
The "Grimm" effect: this permits Grimms like Nick to see through the Wesen's disguise automatically. This effect seems to work for Nick automatically, but only when the Wesen is somehow emotional (upset, angry, scared). There is evidence that Wesen can tell when their disguise slips around a Grimm, but we've also seen Wesen "sense" what Nick is and morph in reaction. Nick himself claims not to know for sure exactly how that works.
"True" shapeshifting: this effect permits the Wesen to intentionally drop their disguise or shift their shape to their true form. Nick has Monroe do this to
show Juliette what he is.
The Grimm effect seems isolated to Grimms, since Hank does not see every Wesen that Nick does, even after he has seen his first and is looking for them. Nick himself believes that Hank is unlikely to see many more Wesen that he already has.
On the other hand, it appears that anyone can see a Wesen that has shape-shifted, which they may do as a way to harness their latent powers. For example, the Coyotl shift before
trying to attack Nick and Hank,
obviously expecting his form to be visible to the nearby humans.
The first time Hank sees this effect, Monroe has shifted form on purpose but runs into Hank in the woods. The second time, Hank sees a killer that has shifted form to try to escape Nick. He does not see his Coyotl friend's true form in the police station, but only when
his daughter shifts form to try an escape Nick.