I used to be a fanatic (heh!) about avoiding overlap. However as of late I've determined this really hinders my game. Points:
- Many cities will never grow to 20 pop to use all the tiles in their fat cross
- 20 pop cities are difficult to achieve before Medicine and Hospitals
- Small cities with unused terrain are wasting space
- Two small cities sharing a lot of tiles are easier to manage from a health and happiness perspective, particularly in the early game
So I've started overlapping all over the place. I do still place with an eye for which tiles are "primary" and "secondary" for any given city, but I have become more productive and am making better use of my terrain. It's also faster to reinforce with the shorter distances.
Just my two gold!
Getting 20+ pop cities is facilitated more by biology than medicine. And supermarkets are better than hospitals since they're cheaper and usually give you +1

and +4

, instead of just +3

assuming you have cow,pig,deer,sheep (I usually have all).
Another thing I'd add is that commerce or GP cities should be built with less or no overlap, and it matters less for production cities. Why? Because the production cities are going to build factories, coal plants, and industrial parks that will limit growth anyway.
In the early game, placing a city that will give you access to what I call complimentary resources is more important. For example, pairing a gold resource with pig, corn, or fish is ideal because you can get huge yields without working a lot of tiles, and you can still work the gold comfortably at 2 pop without halting early growth. If you have to build a city 6 tiles or 3 tiles away in order to achieve this, then do it.
When establishing a city that will overlap with another, I consider the overlapping tiles desert tiles for the purpose of judging the quality of the new city's location.
Also, if you have very limited expansion space for some amount of time. It might make more sense to overlap so that you can utilize every tile you can.