The total linear momentum of a system of particles stays constant provided that

Newton's second law for a system gives $F_{\text{ext}} = \mathrm{d}p/\mathrm{d}t$, so the total momentum only stays constant while the resultant external force is zero. Internal forces between the particles always occur in equal and opposite pairs and cancel in the total, so they never change it. Kinetic energy is a separate quantity: momentum is conserved in an inelastic collision where kinetic energy is not.