How Do I Know That My Puppy Is Infected By Rabies?

If your dog has been appropriately vaccinated for rabies recently contracting rabies is very unlikely.  Rabies in general is very rare in the domestic dog population.  It sounds like your dog was drooling or may have had an accident with very dilute urine.  Your dog should be examined by a veterinarian to discover the source of the fluid-rabies or otherwise.
Rabies is a severe, fatal viral infection.  It enters the body from a bite from an infected animal--fox, raccoon, skunk, and bat are the strains present in the US.  The British Isles, Hawaii, and a few other places are rabies free. 
There are 3 stages to rabies--prodromal, furious, and paralytic.  Initially changes in behavior (anxiety, aggression), odd behavior (wandering, snapping), disorientation and drooling can occur.  Incoordination, seizures and paralysis will occur.  Dogs will die within 7-10 days of showing signs.
There are no good invivo tests for rabies.  If a dog is suspected to have rabies it needs to be placed under an appropriate quarantine.  Your veterinarian will have more information and may be able to quarantine the animal at the clinic.  Healthy, vaccinated animals need to be quarantined for 10 days.  An unvaccinated dog that has been bitten by a known rabid animal should be quarantined for 6 months.  Rules vary by city and state. 
This is a zoonotic (can be transmitted to humans) so if you have some concerns your dog needs to be examined by a veterinarian immediately.

You don't have to worry about your puppy because your puppy don't have rabies.  If your puppy had rabbies then he would be very aggressive.  He would be scared of water.  You will see he won't drink any water and there would be white foam coming out of his mouth and within three to four weeks he would be dead even sooner.  If any animal with rabies scratch or bite you then you need to get to a doctor immediately because after a period of 4 days then nobody can do anything for you but you don't have to worry about your puppy.  You must take the puppy for vaccination and puppies got worms and because they always lick and bite you then you can get them to so you must have the puppy de-wormed and at the pharmacy you can by pills for the whole family to drink so you don't get the worms.
 

If these dogs had been previously vaccinated with rabies then it is highly unlikely that either dog has rabies and/or would have passed it on.  If there is any concern regarding human health especially of a child seek medical attention immediately.  Rabies is 99% fatal especially in children.  If rabies is a true concern these dogs may need to be sacrificed for testing.  The only way to definitively diagnose rabies is from a brain tissue sample.
Work with your veterinarian and pediatrician to help your son.
Most of the rabies in the United States is found in wildlife reservoirs--bats, raccoons, foxes, and skunks carry the 4 different strains.  Any warm blooded mammal can have and transmit rabies.  Overall prevalence is low though there are some endemic pockets.
Clinical signs occur within 10 days of exposure and all infected dogs die within 7-10 days of clinical signs.  Signs include changes in attitude (animals become shy, nervous, or aggressive), erratic behavior (wandering, roaming, biting/chewing compulsively), disorientation, incoordination, seizures, paralysis, and excessive drooling.
A healthy, vaccinated dog should be quarantined for 10 days.  Contact your veterinarian to find out what your state's requirements are for rabies suspects.

Well if your dog gets aggressive and you can see foam and also if it's walking weird

Follow the advice from the vet lady and in the meantime, watch CUJO and hope this doesn't happen to you!!

It is required by law to have dogs vaccinated for rabies.  This is a serious, fatal disease that can affect humans.  Once a dog is infected with rabies it will progress through a series of stages and end with death.  This typically occurs within 10 days of exposure.  It is highly unlikely that your dog has rabies but until it is vaccinated has the potential to contract this disease.  In most communities there are low cost vaccine clinics or shelters that will vaccinate pets. 
Biting is an unacceptable behavior and you need to be proactive and take steps to stop this.  Tell the puppy "no bites" give it an appropriate toy then leave it alone.  Too much negative attention can become a game and will not help.  Reprimand once and do not continue to give more attention.  Other method is to make biting uncomfortable for the puppy.  Place your thumb under the tongue and hold the lower jaw for 10 seconds.  Do not hurt this puppy just make it uncomfortable and the biting behavior will stop.

My Little jack russell had rabies I think and we had him put down he was doing all the symptoms but he had been in a kennel and not out my boxer however runs lose and I gave him a rabies shot the day that this other dog got put down and he seems to be fine is he okay to be let go I have him on a chain.. Hes been on it for two days now.