I remember in the 1980's Mini-Series Legends, which I think predates Wally's Flash series, he was trying to figure out if he should pretend to be Barry or let the world know he was the new Flash and if he should have a secret identity or not.
Wally West was fighting Dr. Alchemy after Barry's Death as Kid Flash when Dr. Alchemy turned his suit into vapor, making him naked and revealing that he was Wally West to the world. Both Superman and Jay grilled him on making his identity public in such a foolish/amateurish way. Wally had been reckless because he was angry about Barry's death and was afraid to take on the mantle of the Flash. Wally tries to convince Jay to take it back but Jay just goes after Dr. Alchemy alone. Wally ends up saving the day using one of Barry's old Flash Costumes (because his Kid Flash costume was already torn to bits).
It was okay for the public to know Wally's identity because there was no one that would be hurt by it. Barry's parents died from shock when they found out that Barry had died and Iris was "dead", so Wally and Jay revealed Barry's identity to the world. Jay's identity was already known to the public when he retired in the 1960s. Wally's immediate parents might have been harmed with his identity made public, but I think Rudolph's connection to the Manhunters might have kept the Flash Rogues from going after them (also, going after family is against the Rogues' code).