I live in Cleveland and work in Beachwood, OH. I wasn’t prompted to send info to either for my city income tax W-2. What do I do now? Is there something I should do?

TurboTax (TT) does not automatically prepare a city return, you must initiate it. At the screen, in the Ohio program, titled 
 "A few things before we wrap up your state taxes"
      scroll down to Ohio City taxes. 
Or, later
"Your 2016 Ohio Taxes are ready for us to check"
    -Click Other situations
    -Scroll down to Ohio City taxes. 

Cleveland is a CCA member city, so TT can prepare your city return. TT cannot e-file an Ohio City  return. It must be printed and mailed.

I actually find it easier to fill out a city return by hand or at the city, CCA or RITA web site, than to use TT. Things don't happen automatically and you need to know what you’re doing. The actual  RITA/city forms are easier to follow than the TT interview.

http://www.ccatax.ci.cleveland.oh.us/

 You normally only have to file a return in the city you live in. It is not necessary to file one in the city you work in, if different. Your employer will remit withholding to the city you work in, but you do not file a return unless you're claiming a refund or your employer didn't withhold enough ( both highly unusual). The city you live in will allow a credit or partial credit for the withheld tax you paid to the work location city. 

That’s crazy, because it isn’t even blatantly obvious that you have to do that. It’s just so confusing. Thanks.