Question: There were no mirrors inside the Chamber Of Secrets. If there was one, would the Basilisk get petrified if it saw itself in the mirror?

On a more complex point, what would happen if two Basilisks made eye contact?

**Edit:** The water in the Chamber Of Secrets should be neglected. If a snake moves in water, it sends out ripples which make the image of the Basilisk unclear for it to see itself.

Answer: 
As far as I know, no canon info on this, as there is only one basilisk in all of the seven books, so there is no example of an altercation between two of them. 

If I had to take a guess, I don't think a basilisk would be petrified by its own reflection. 

The basilisk had to enter the castle and leave through the secret entrance in Myrtle's bathroom, where there is liable to be a number of mirrors and puddles of water, so it is unlikely that the basilisk never accidentally caught a glimpse of itself. 

We know form canon that an indirect look into the basilisk eyes (through camera lens, Nick, mirror reflection, etc) only reduces the potency of the basilisk stare, but still causes some damage (that is, petrification). As in the Harry Potter Universe, certain powerfully magic creatures are immune to a certain level of magic (eg, dragon and giant skin), and the basilisk being a rare and powerful magic creature, the reduced level of potency of it's stare in case of indirect eye contact may not be enough to cause damage. 

As for what happens if a basilisk looks into the eyes of another, there is no way to tell based on the 7 books (nor Fantastic Beasts, as far as I know).
I think a basilisk is immune to the stare of another one, just like most venomous spiders, scorpions and snakes are immune to the poison of their own kind.