You have put your finger on one of the mysteries of quantum mechanics. The experiment you ask about is essentially the same as one that Einstein and collaborators wondered about in the 1930's.
Let's say that you first measure the position of particle A to good accuracy. That puts A in a state with a broad range of possible momenta. Now you measure B's momentum to good accuracy. That can tell you what momentum A had back before you changed A
by measuring its position. So the combined measurements certainly do not tell you the current
momentum-position combination of anything to better accuracy than allowed by the uncertainty principle.
In fact, you cannot use these measurements to assign overly accurate x-p combinations at any
time, even the past. Whatever quantum states described A always obeyed the uncertainty principle, by mathematical necessity.
Now if you're as subtle a thinker as you seem to be, your next question might be "Yes, but how do I know that any quantum states provide a complete description? Maybe particles have some other properties, beyond those in the quantum states." That's what Einstein thought.
Nothing we do can show that there are no properties at all beyond the quantum state. However, a type of subtle experiment motivated by John Bell can show that no properties of the type we might like for particles to have (e.g. defined position and momentum of individual particles) can exist. These experiments have now repeatedly been performed in a variety of systems. The results are consistent with the idea that there is nothing but quantum states, and, more importantly, inconsistent with the idea that nature has all the definite properties that we feel it should have. See, e.g. http://en.wikipedia.org/wiki/Bell%27s_theorem
(published on 04/17/2010)