Firstly, make sure you have Xtal2 (pin 4) of the master oscillator connected to Xtal1 (pin 5) of the slave oscillator. The capacitors are on the master oscilator and Xtal2 is not connected on the slave oscillator.
What are the specifications of the crystal? Load capacitance, package shape etc. How far apart are the two microcontrollers?
The capacitors are there to provide stable operating conditions for the crystal. C1 provides load so that current can flow in the crystal and C2 helps to smooth the fast edges of the inverter switching. Without suitable conditions the crystal won't oscillate. In your case I suspect that noise from the power supply caused unwanted oscillation which prevented the crystal oscillator from starting. The capacitor reduces the noise and allows the oscillator to work normally.
The crystal will take up to about 1 ms to start up. At start-up, wide-band noise is being amplified by the oscillator amplifier, and the crystal only allows through the correct frequency, which gains amplitude until the amplifier saturates. If you look at Xtal1 or Xtal2, with an oscilloscope, there will be no visible signal for more than half of the start-up time, before the signal amplitude increases exponentially. During the early part of start-up, the circuit is very sensitive to noise.
This shows a typical crystal oscillator start-up:-
You can get an unwanted oscillation that starts up much faster and prevents normal crystal operation. You can even get the situation where if you touch the crystal connections with a 'scope lead, the oscillator starts and is then stable with the scope lead removed.
The correct value for the capacitors is whatever gives the most stable operation.
Changing the capacitor value will also change the frequency a bit. If you are not worried about 0.1% or smaller frequency errors, you can ignore that. If you are, then you may need to adjust the capacitor values to give the correct frequency, and it will probably not affect the operation noticeably. If it does, you need to buy a crystal with a different load capacitance, or change the circuit design.