It probably depends on what Deathwatch you are referring to.
There's
GW's version, which is an integrated part of the Inquisition's Ordo Malleus, and then there is the interpretation that FFW came up with for its series of
40k RPGs, which is an independent entity merely allied with the Inquisition. There's probably lots of novel author interpretations falling in-between the two, too.
I'm more familiar with
GW's original version, so I'll try to add something from this direction; I'm sure other dakkanauts can fill in with other, equally valid accounts from the
RPG or the novels.
It seems to me that the Chapters sworn to assist Deathwatch are conducting their own selection, basically entering the names of suitable Marines into a Chapter's roster of personnel suggested for deployment in the
DW. An example would be Captain Artemis, whose background can still be accessed on
this archived website.
Inquisitors likely have too much to do to "waste time" observing Space Marines - however, I'd deem it entirely possible that an Ordo Xenos Inquisitor who has secured the support of a Chapter in an anti-alien operation may, by pure chance, become convinced that a particular Space Marine would be suitable for this job, and so may request or requisition him from the Chapter Master, either by inducting him into his or her own team, or by suggesting to send this capable individual to the Deathwatch.