In meetings with my teammates from USA they pronounce my russian name "Artem" with American accent(not the way other russians pronounce it). Should I also pronounce other's Russian names with the accent or say them as I usually say in Russian?
For your own name, you decide how you want Americans to pronounce it (but be merciful).
Other's Russian names might be unpronounceable to Americans, so you need to find out how those names are usually pronounced in America.
Without crossing the Atlantic, or even travelling to Russia, listen to how the composer Mozart's name is pronounced in France. The point is that it can sound very pretentious for an American or a Brit to pronounce a foreign name absolutely correctly as if speaking that language, but it can sound horribly ignorant to pronounce a foreign name as if it were English. There has to be a middle way, but it is not always easy to find it.