Solzhen is right (and gives some good examples), but I would add that, at least in British English, "leave it with me" can also mean virtually the same as "leave it to me".
"I can't get my website working properly."
"Leave it with me. I'll sort it out."
(The meaning is "leave the problem or the job with me".)
I have no idea which is more common. Sorry.
"We are not wholly bad or good, who live our lives under Milk Wood"
prayer of the Reverend Eli Jenkins in Under Milk Wood by Dylan Thomas