The common vernacular in English is "take a shit" though I've never felt the urge to take any shit. On the other hand I leave shits all the time. Would you rather take a shit or leave a shit?
The phrase "take a shit" is using shit in its verb form e.g. I gotta shit. To me "leaving a shit" feels like you just physically have a piece of shit on you, in your hand or in your pocket and just leave out on the counter or something.