Assassin only protects you from showing up to UAVs and similar "radar" sourcing devices when you are not shooting. Only a silencer conceals you when shooting.
You can see from this link:
Makes you undetectable by UAV, thermal scopes, Heartbeat Sensors and Portable Radars. You can still show up on the radar if you fire a weapon that doesn't have a silencer.
Note also that Assassin is not effective against Recon or the Recon Drone, although in my experience you do not get a targeting reticle on enemies using Assassin with the Recon drone, even though you can still tag them if you see them.
Assassin doesn't protect you from the Recon perk or the Recon Drone killstreak.