It really varied from military to military and was highly dependent upon what type of equipment was captured. Highly advanced technical equipment like new aircraft models were almost universally preserved by all sides and sent back well behind the lines for study of their technical secrets. Captured new tank models often became highly-valued war prizes and were not wasted being used against their former masters. For example, the Soviets captured a Tiger I on the Leningrad front in January 1943 and immediately sent it back well behind the lines for evaluation. In places where logistical resupply was quite difficult, using captured equipment was one expedient solution. In North Africa, the British, Germans, and Italians each used each other's captured equipment. The Soviets occasionally used Tigers and Panthers n the front, but discovered they frequently broke down.

The Germans most frequently employed captured equipment in combat, often designating them Beute models. This was in no small part due to the shortcomings of German industry and the general unreliability of some German equipment like the fabled Panthers and Tigers. Additionally, the Germans needed captured cars and other motor transport to keep large formations mechanized. The Germans had captured large stocks of Allied, especially Soviet, equipment between 1939-42, and that allowed them to escape the common problem of using captured stocks: ammunition and spare parts.

The only piece of Axis equipment regularly used by American forces, the Panzerfaust, demonstrates why other Axis equipment was not used regularly. The Panzerfaust was a man-portable anti-tank weapon that was easier to use than its Allied equivalent, the bazooka, and much more portable. The Panzerfaust was also better than the Bazooka against hardened bunkers and could be easily stowed in a jeep for emergencies. Its one-shot use meant that the issue of resupply was not an issue. The one-shot and small nature also meant that it mitigated the issue of friendly fire as it was concealable and its audio-signature was not repeated once fired, unlike, say, a captured machine gun. In short, the Panzerfaust did its job slightly better than Allied equivalents and its one-shot design mitigated the usual problems associated with using foreign equipment. This was not the case for more advanced equipment like tanks.

But in man ways, situations like North Africa and Germany were the exception rather than the rule. Using enemy equipment compounded already complicated logistical problems and even more dangerously, greatly increased the possibility of friendly-fire accidents. The German's use of Beute equipment was less a matter of military efficiency and more desperation on their part.

If the situation permitted then the vehicle was employed in the service of the army that captured them. The Australian's used the Carro Armatto tank with a white kangaroo painted on them for help with identification.

http://ww2photo.se/tanks/aus/utland/06779.jpg

Many Luftwaffe aircraft that got captured during the war were test flown at RAE(Royal Aircraft Establishment) Farnborough. These aircraft got allied markings painted on them, so as to avoid misidentification by friendly fighters over Britain. I have not heard about Luftwaffe aircraft being employed in combat duty in allied service or the other way around.

I'm pretty positive I've also read a post in this sub concerning the equipment the Fin's used during WWII, which consisted of a wide variety of equipment including captured from the Soviets. Unfortunately I did not save the post, and the reddit search is horrible once more.

Follow-up question: What did the general public think about their armies using captured vehicles? What did the soldiers think?