These being two distinct phrases. Using the definitions found on Wiktionary:
let's see: used as a filled pause to indicating thinking or pondering, but allowing hearer to participate
let me see: used as a filled pause to indicating thinking or pondering, but not inviting hearer to participate
How do I indicate the fact that I'm thinking about something? And how might I imply "if you have an idea/thought, share it" or 'dis'-imply it?