Excuse me if this sounds ignorant as I am not a security expert, but isn't there a flaw in using a public key for continuous messaging? Shouldn't public-private

Excuse me if this sounds ignorant as I am not a security expert, but isn't there a flaw in using a public key for continuous messaging? Shouldn't public-private key crypto be used only to generate a symmetrical key? The system was originally designed for symmetrical key exchange no? Using it this way presents some flaw?