ok...as a wildlife rehabber, this is a situation i get called on frequently...
hit wildlife...we also get alot of hit pet calls
this is my take on the situation:
In town:
widlife: staying in your car see if you can see the animal, is it still moving. wildlife when injured and corenerd will attack if they have the strength...
if the animal is still moving...call the cops, report the hit, you can either stay with the animal or not...
but all that will happen is 1 of 2 things, the cop will have a rehab person like myself come out to asses the animal, the animal will then be either taken by the rehabber or imediatly ethanized either by the officer via gunshot, or by the rehabber typically by snapping the neck. 9sounds horrible but this is the most swift and painless way of euthanizing an animal when you dont have drugs, a gun or a knife on you. if the cop doesnt call a rehabber the animal will typically be shot.
please note you will wait for a while in most towns for the cop to arrive, its not an emergency afterall.
if its a PET, again you want to use caution animals in pain may not react normally, agression is not uncommon in an animal in pain.
i would get out, if the animal is dead with no identification it would get pulled off to the side of the road to prevent smushing and notices woudl be posted locally describing the animal, where the incident happend ect.
if the animals still alive it would depend on the condition of the animal, im trained in animal first aid and emergency responce so if the animal isnt to the point of nothing but a twitching corpse i would attempt to stabalize before loading up to take it to the nearest vet.
if the animal is close to death with little hope of surviving...i will use the same technique as is used on wildlife beyond help. either by hand or using a large rock, snap the neck.
sounds barbaric but suffering is worse.
again i would make all attempts to make sure the owners of the animal were made aware if possible.
in the middle of nowhere...
widllife:
either keep going, or pull over to check it out...most rural cops wont come out unless its smack damn in the middle of the road...
personally if the animal is aware and seems ot be in good shape ill leave it...
if the animal is neer dead, a quick neck snap and then depending on what it is and how badly its been damaged itll get picked up, i know lots of raw feeders and work with a raptor rehab center, road kill rabbit (and deer) are highly sought after...(though deer are too big for me to pick up.)
otherwise ill dragg it off to the side of the road so its out of the way when the scavangers come around and be done.
if its a pet: same rules, if i can and can get it to a vet in reasonable time ill stabalize and take to a vet...it the vets too far away i will often pick up, stabalize and try to keep hold of it till i can get it to a vet... but if the animal isnt going to make it despite efforts neck snap...
i cant in all good knowledge allow an animal to suffer, id rather get my hands dirty and finnish the animal off than leave it there to duffer (or to blead to death in the back of my car...)
and as other posters have said, unless you are trained to do so NEVER EVER EVER pick up a wild animal thats been hit. in the spring if you suspect theres babies feel free to look for them box em up and contact a wildlife rehabilitator but NEVER touch a wild animal especially if injured!