Water leaking from back of engine on passenger side

radiator hose or rusted out welsh plug
get a coolant system pressure test done to determine the exact location of the leak

You may have a plugged AC water condensation drain hole, it drains under the car at the base of the engine bulkhead on the passenger's side.

It sure sounds like the heater core, especially if you have water leaking inside the cabin. The two heater hoses only go from the engine to the firewall, and do not go into or through the firewall. They connect to the heater core right at the firewall.
Check carefully around the back side of the engine and the heater hoses, though. If a freeze plug is leaking, water will be dripping off the bottom of the engine. If it is leaking before the firewall, that would not be the heater core.

this could be from broke or busted heater hose going into fire wall at back of engine on passenger side or clamps could be loose causing leak in this area

indicates a rusted out welsh plug
take it in for a cooling system pressure test

Your heater core has developed a leak - the wetness on the passenger side carpet is engine coolant
from a heater core leak, or a leak in one of the two hoses connected to the back of the heater core.
Your strategy: disassemble dash down to heater core, determine whether you'll need a new heater
core or new hose (or new hose clamp), then replace the offending part.

is it coming from around that back of the engine on passengers side?? if so you probably have a water pump thats bad, and starting to leak..

The water pump is on the passenger side of the engine below the tensioner. Also look for coolant leaks from the head gasket.

there are core plugs in the back of the engine block....one is rusted out....you will need to remove the trans to replace them