If these encounters did occur, what were they like? Did the opposing sailors ignore each other? Help each other? Did theyFight??
While it isn't exactly what you are looking for, one situation that is similar to the topic of your question came during the Battle of the Heligoland Bight, in the first months of WWI. The battle was a British attempt to strike German patrols in the Heligoland Bight using light forces (destroyers and cruisers) supported by battlecruisers under Admiral David Beatty. The early stages of the battle saw the British light units tangling with a smaller force of German destroyers. During this engagement, the German torpedo boat (small destroyer) V-187 was hit heavily, and began to sink, assisted by scuttling charges laid by her crew. A flotilla of British destroyers approached the stricken ship, and Defender lowered her boats to take off the crew. However, just eight minutes after this had begun, the appearance German light cruiser Stettin forced the British ships to withdraw.
Defender' was unable to recover two of her boats, abandoning their crews to what seemed like certain captivity. Stettin declined to pick up the boats, first steering to engage the British destroyers, and then withdrawing to repair damage sustained in the engagement with them. The crew of Defender's boats took this abandonment in their stride, bringing a considerable number of German survivors aboard. A total of 28 Germans (18 wounded, ten unwounded) were taken aboard the boats, which were crewed by just ten Britons between the two. The German survivors were well treated, with the wounded being looked after well. Fortunately for both the British and Germans alike, they were being observed by the British submarine E-4, commanded by Lieutenant Ernest Leir. Once the coast was clear, Leir took E-4 to the surface, and approached the boats. Leir took the British seamen aboard the submarine, along with three injured Germans. He left the rest of the wounded in the boats, along with the unwounded men, giving them food, water, and a compass with which they could steer their way back to German-held territory.