It might help if we ever witnessed crime on Earth. That is, crime not committed by a Starfleet member. Or even a crime committed at a location where Starfleet by default isn't the only government authority within a dozen lightyears.
As matters stand, the only police authority ever witnessed in the Federation is Starfleet. It's Starfleet that jails McCoy in ST3, Starfleet that persecutes Tom and alt-Harry in "Non Sequitur", and indeed (United Earth) Starfleet that investigates the kidnapping of Dr Phlox already, perhaps setting a precedent to how things will be done once the Federation is founded.
Entrusting law enforcement on the military is by no means uncommon, even in so-called free countries. Various militia- or gendarmerie-type setups exist all across the globe, the only real difference between them and "civilian" police being that the ministry/department in charge is that of Interior rather than Defense.
I could easily see the UFP streamlining things a bit and moving police duties to under the Starfleet umbrella, perhaps along with customs, firefighting and municipal gardening. For all the implications of a socialist paradise, the UFP might believe in a very sharp rift between private business and social-communal services, leaving the latter utterly to the state.
As for how the Earth police look like, never mind which organization they serve under... I posit that the role of the beat cop has ceased to exist by the 24th century, and that it has been deemed that visible police presence is undesirable in every respect. The police will beam in only when needed.
Timo Saloniemi