They can definitely be stable with respect to one another. In fact, there are several possible configurations you can put two moons in order to gain this sort of stability. Here are a couple of them:
1: Moons in separate orbits
The trivial solution to this is to put your moons in separate orbits far enough apart that they don't interfere with one another. The moons around the gas giants mostly behave like this.
2: Moons form a binary pair
If your moons are significantly closer to one another than they are to your planet, they're likely to form a stable binary pair, orbiting their common center of gravity. These moons would likely be tidally locked with each other, rather than the planet. They'd also likely need to be further from their parent planet than Luna is from Earth, since if they orbited too closely, the gravity of their parent planet would likely throw off their orbital pattern.
3: Moons form a horseshoe orbit relative to one another
It's also possible for similarly sized bodies to stably orbit a planet while sharing an orbit if they form a horseshoe orbit with one another. Epimethius and Janus, two moons of Saturn, are good examples of this sort of system. From the surface of the planet, your moons would appear to vary in distance over time, from a position fairly close to one another in the sky to one in which they were on opposite sides of the sky. They'd oscillate periodically between these extremes without ever touching one another. They'd also vary in size, taking turns appearing to be the larger celestial body. This would no doubt lead to lots of interesting stories in the mythology of your planet dwellers.
4: Moons which sit in each other's Lagrange points
Specifically, we're talking about the $L_4$ and $L_5$ Lagrange points. These points are dynamically stable, so barring any particularly large perturbations, two moons should be able to sit in those points relative to one another. These systems are less stable against perturbations when the bodies are similarly sized to one another, so if there are other planets in the solar system, one of the moons may be ejected from its orbit after a period of time.