Any measurement device used to witness the phenomenom seems like it would disrupt quantum entanglement. How do scientists manage to observe it?
True, but the statistics after a measurement are very different from those you expect in classical physics which does not allow entanglement. The various Bell inequalities are ways of testing the theory. The simplest one is CHSH where the chances of winning a certain game should be 1/2 but are sqrt(2)/2 for entangled pairs. This can be tested. Other tests are protocols such as teleportation which cannot work without entanglement . One of the big experimental problems with entanglement is that it is very hard to directly check if you have an entangled state until you actually use up the entanglement. (and even then it's usually a problem for a single system). If you do have some other entanglement which you can "trust" it is possible to make a non-local measurement such as a bell measurement (not to be confused with the bell inequality) to verify if your system is entangled, this is very hard to do in experiment. What you need for that is an entangled measuring device which lets you record the modular sum of the spin in the X and Z directions.
Fwiffo gave you a very good answer. Maybe the simplest example would be if you send 100 pairs of photons thru two parallel aligned polarizers. If the photon pairs are entangled – you will have 100% correlation, i.e. 100 measurements of (1, 0) or (0, 1) (i.e. if one photon get thru the other gets stopped). If the photon pairs are not entangled – you will have 100 random measurements of (0, 0) or (1, 1) or (1, 0) or (0, 1). (Note: To get real violation of Bell Inequalities, you need to do more measurements on all relative angles, not only parallel.) P.S. There is something called Entanglement swapping where Quantum teleportation is used to entangle particles that never interacted with each other! :surprised