They can do that, since they own the NFL Network and ESPN is their broadcast partner for the event.
Other networks, however, have completely free reign over what they can and can't tweet about during the draft. CBS (and former NFL Network) football reporter Jason La Canfora has announced that he'll be tweeting out information as soon as he learns it, and that CBS will be reporting information as quickly and accurately as possible.
So keep this in mind. If you're following certain guys (like La Canfora), you'll learn about some draft picks before they're actually announced on television. So if you're into that, then pay attention to his Twitter feed.
On the other side of the spectrum, some people don't want that. So the answer is simple.
Various football reporters will be notifying people about whether or not they'll be tweeting out picks before they're announced. If you want to hear it, follow them. If you don't, maybe you should stay off Twitter during the draft.
Don't say I didn't warn you.