On the contrary, this is evil genius. Once the rumour spread that happens, people stop communicating about the war.
At the human level there are probably plenty of low-tech techniques to keep yourself safe like deleting messages or using coded language.
99.9% of the population wont take any risks (e.g. even receiving text messages about the war). They will closely monitor their phone and block anybody who dont do the same.
Fear, as you say, is the purpose.
I would add that multiple aspects of Putin's attack plan for Ukraine relied heavily on its ability to instill fear.
Putin's regime expected Zelenski to flee at the sight of Russia's military at the border, they expected border cities to promptly fold at the sight of Russia's army, they expected Ukraine's army to surrender once Russia's army gained some ground, they expected all NATO and neighboring countries to stay quiet and did nothing when threatened, implicitly or explicitly, by them.
But thankfully their bluff was called, and Putin's regime was exposed as the fascist paper tiger they are.