The fancy thing about matter streams is that once they are out of the sending machine, they seem to be quite survivable. In DS9 "Dramatis Personae", a Klingon pattern starts to rematerialize, some seconds after the sending machinery is destroyed in an explosion. It does materialize on a receiving platform, and the receiving machinery then takes over, so that what was originally a red Klingon pattern is turned into an amber Cardassian one before becoming solid matter. But it still seems that the matter streams are fire-and-forget things.
In light of that, it seems the stream was sent into the buffers or other innards of the Enterprise
, rather than simply being sent into the middle of the transporter room and allowed to rematerialize there on its own. Which was a bad choice when the receiving machinery wasn't working properly, but might have been an advantageous idea otherwise, because from "Realm of Fear" we know that locking two transporters together (and essentially sharing their pattern buffer contents) gives better range and/or penetration in difficult circumstances.
If what we're seeing is "Realm of Fear" type sharing, then the pattern might be in one of the machines, or partially in both, and moved like a hot potato - but since one machine was out of order, the pattern would be suffering whenever it was in the bowels of that broken machine. With two well-working machines, the two transporter operators could have played tennis with pattern basically forever...