In Ontario, this is what our Provincial Interpreter has decided and it makes sense to me.

Further to what is provided in the Federation Manual,
the official who calls the time out will report the time out.
The other official will be at the point of the throw in with the ball. If the throw in is in front of the team bench, the official will be out on the floor, in line with the reporting official, as has been done in the past.

At the conclusion of the time out, the official who is at the point of the throw in, will administer the throw in. The official who was at the table will move into the proper position to cover the throw in. This may necessitate a switch from the positions the officials were in, prior to the time out being called.

It's all random choice. But that's my point -- do whatever your local powers to be want.

We didn't go over any proper mechanic. This is however what all officials do. The calling official will report the timeout get in place, and the partner will either stand at the throw in spot or but the ball down at the throw in spot and stand with his partner in the designated spots.

I'm lost on two man. Had a GV last night two man, terrible teams, small, very private schools, nobody in gym. Partner calls timeout, reports. I stay at spot on endline. First horn, and I think what the heck, I'll bring out home team as they were right there. I put ball down and it brushes my shoe and starts a slow roll. Not to worry, as it always comes back. But not this time. I turn around and can't find it..........gone. It had rolled under stands. I break my cardinal rule and try and find it. can't squeeze between wall and stands. Home player goes under stands and comes out with TWO balls. Little chuckle and she gets a golf clap from the crowd of 10. Later in quarter, another timeout and partner tosses ball my way as he reports TO. I don't see him toss ball and same thing happens! Same girl goes into catacombs to retrieve it again. That was a microcosm of my game. GIGO. I was as bad as the teams were.:o

Note, however, that this is NOT the Federation mechanic. A lot of areas do it, and I find nothing particularly wrong about it, but it is not the NFHS method.

NFHS mechanic:

The official who will go to the throw-in spot is determined as if a defensive violation occured at the spot of the ball. It doesn't matter who grants the timeout. If the post-timeout throwin is on the FC endline or the pre-timeout lead's FC sideline, the pre-timeout lead will be the one marking the spot. Otherwise, the pre-timeout trail will be marking the spot. The official not marking the spot goes to the division line. If the throw-in spot is above the FT line extended but still in the frontcourt on the pre-timeout lead's side, it will casue a switch....the same as if a violation had occurred in the same area.