For example, could Captain Picard command the computer to use half of Holodeck 2 for his Dixon Hill program, so that Worf can use the other half to fight monsters?
I suppose it's possible. Not only two programs, maybe even more. Nevertheless, as Cadence stated, they aren't that big, so that it could be possible to accidentally touch or hit another person.
But: If you watch the episodes with long running programs (This village in Voyager, Vic Fontaine in DS etc) the same holodeck is used for dozens of crewmen. They are not always in the same room or place within the program. This has to be handled by the computer.
If you image the size of Vic Fontaine's club, it's not even close to the size of the holodock in Quark's. So if some crewman is at the bar and another one near the stage, it must be virtualized by the holodeck.
The holodeck program has also to care about noise and sounds. The one at the stage hears music much louder than the one at the bar. Also, they should not hear the dialogs from each other.
So, I think eventually it is possible in theory, but the computer of the holodeck must also have the needs for that (running two or more programs simultaneously could be a challenging task)
@christoph's answer is perfectly good.
Consider also that we know it's possible to run a holodeck simulation INSIDE a holodeck simulation. Like when Picard realized that he was inside a recreation of the Enterprise that Moriarty created. (TNG: Ship in a Bottle). The "exit" from 221B Baker Street to the Enterprise hallway was just a door.
So, to answer your question, yes. You just have an "exit" from Dix's office to some jungle full of monsters. Then "close" the "exit." Easy as pie.