GP points are specific to a city. The only circumstances where you get GP point "conflicts" are Great Engineer, Great Merchant and Great Scientist points, since all three types of GPs operate on the same counter (i.e., generating a Great Engineer increases the cost of the next GE, GM and GS by 100 points). But you need to take care with that whether you are running GE, GM and GS specialists in the same city or different cities. All other GPs run on separate counters, so you can run them in the same city or different cities, as you see fit.
Filling specialist slots (which you should do manually -- don't rely on the automatic city governor) does require a source of excess food to support the specialists. If you run too many specialists in a city, you will slow down population growth in that city. In addition to building and working farms, building food buildings, and befriending and allying maritime city-states, internal food trade routes can supply extra food. If you adopt the Freedom ideology, there is a Level 1 tenet, Civil Society, that will cut specialist food consumption in half.