Not silently. There's a voice alert, "Priority Left" or "Priority Right", and a red light. But with other alarms active, it's likely to be missed.
Airbus statement on sidesticks. [1]
Boeing position on sidesticks.[2]
This is a long-standing controversy. The F-16 fighter was the first aircraft produced with a sidestick. But that was so the pilot could control it during high-G maneuvers. Airbus's use of it has been controversial for years. For a good overview of the Airbus control system, read "Fly by Wire", by William Langewiesche.
But that wasn't the problem with Lion Air.
Airspeed. Attitude. Vertical speed. Altitude. If the pilot is operating on wrong information for any of those, they may lose control of the aircraft. All large aircraft have more than one of each of those. But when they disagree, or fail in a common mode, big trouble.
Whether some other data source like GPS or inertial should be used to back up those is an open question. GPS can and is jammed. Inertial systems drift. Both are complicated. The classic instruments are so simple. But they rely on tubes to holes in the aircraft skin, which might become plugged for a number of reasons - ice, maintenance covers, insects, or dirt.
[1] https://safetyfirst.airbus.com/app/themes/mh_newsdesk/docume...
[2] http://archive.li/x0Pn8