Eye discharge, nasal discharge, sneezing, laboured breathing.  Lack of appetite. Lethargy.  Elevated temperature.

There are a few things to note  - you can get both upper (affecting the nares/sinuses) and lower (affecting the trachea and/or lungs) respiratory infections.

Symptoms of Upper Infection
Temperature (signs - hot, flushed ears)
Nasal discharge
Excessive sneezing
Poor appetite
Snuffling/noisy breathing
In advanced cases the chinchilla may start to "mouth-breathe"

Symptoms of a Lower Infection
Temperature
Laboured breathing (in serious cases they may even start coughing)
Rattling/wheezing noises in chest
Poor/no appetite
Generally off-colour and lethargic

Possible Causes

Colds can be contagious.  A damp, draughty environment can also trigger respiratory infections as can a poorly ventilated, dusty environment. Pneumonia can be viral, bacterial or have parasitic causes.  Any form of respiratory infection needs prompt veterinary treatment.