Discord has announced that they have just starting testing video chat and screen sharing for 5% of desktop players, and this test is to iron out any bugs, crashes, and glitches, as well as seeing how the servers handle the increase in bandwidth.
Regardless, if you're one of the lucky few who got access to the new features, you'll be able to start video calls with anybody, although be aware that Discord may disable video calls suddenly depending on how the test is going.
In regards to a full rollout of the feature, Discord says "We're still at least a month out from considering a full launch. Why? We're measuring and testing bandwidth; acquiring and implementing more backend servers; and praying our code actually works. This takes time. It could also be longer than a month."
Could this rival Skype?