Generally it is the principal's responsibility to make sure all the parts have the correct bowing. In student orchestra's the principal tends to do everything him/herself.
In professional orchestra's, if bowings are changed, the principal does it in his/her part and then the orchestra librarian copies them over. Many times professional orchestras have standardized bowings, as well as many schools. A great example of this is the opening passage in the bass part of Don Juan. When I was in NY, the NY Phil started it up bow, and so did we at Juilliard. But my friends over at MSM started it down bow.
A lot of times as the principal, I try to learn my part completely with several different bowing variations and then set up a meeting with the conductor and other principals. Here we will discuss bowings, play along with the conductor at the piano (or sometimes the conductor just listens), to create uniform bowings in tutti sections and to also make sure it is what the conductor is looking for.
Some basic rules though are that pickups are almost always started with an up bow unless specified by the conductor. Also, depending on the piece and composer dotted 8th, 16th note patterns are either hooked or played with a "shoeshine" stroke. The shoeshine stroke is pretty standard in the famous passage in the 4th movement of Berlioz's Symphonie Fantastique, while hooking is usually a standard in Mozart symphonies.
This is only just a beginning of an introduction as far as bowings go. I can, as well as many other players can go on for hours about this. The only way to pick all this stuff up is experience and playing pieces several times with several different ensembles.
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