In PIV, the suffix -isto is primarily defined as:
Persono, kiu profesie, daŭre, prefere aŭ ofte sin okupas pri la afero difinita de la radiko
Person, who professionally, continually, preferentially, or frequently, is occupied with the subject defined by the root
The root-word does not have to be a verb or to possess a verbal form in routine use: consider maristo or arbaristo for example. (Note however that Esperanto permits extremely free verbification, and polici, mari and arbari are valid words. Polici is included in Wells's dictionary.) PIV mentions policano once but then defines kaŝpolicisto as plain-clothes policeman.
The word policisto appears in Fundamenta Krestomatio, Marta, and Metropoliteno. The word policano appears in the Ekzercaro and La Revizoro. Both appear in La Faraono and both are used very frequently in Monato articles.
If you need to refer to a person who studies police forces academically you could use policologo.
As for the difference between -isto and -anto, PIV says:
La suf. ist montras precize la profesiecon kontraste al ant, kiu signifas fojan, okazan faranton, k al ul, kiu montras iel personiĝon de la agado: «trompanto» fojfoje trompas, «trompisto» vivas el trompoj, «trompulo» estas homo esence karakterizebla per trompemo.
The suffix -ist- indicates precisely the professional nature in contrast to -ant-, which means doer from time to time or on a specific occasion, and to -ul-, which indicates a sort of personification of the action: a trompanto deceives people from time to time, a trompisto lives by deceiving people, a trompulo is a person essentially characterized by a tendency to deceive.
The verbification of the root-word polico follows the pattern of intended purpose: just as the intended purpose of a broso is to brosi, the intended purpose of a polico is to polici.
In nearly all contexts there should be no real difference between policisto and policano. Some people might consider use of -isto to be wrong for a volunteer or temporary member of the force, but that seems questionable to me.