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Should proving key be kept as a secret in NIZK?

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Hi I am quite new to NIZK. I know a trusted party (Generator) generates proving key and verifying key and then distributes them to Prover and Verifier. Apparently, the verifying key can be seen as public for anyone in the system, but what about the proving key? If the proving key is known to a different party other than the original prover, is there any problem such as another party can forge proofs, etc? Also, if the proving key can be set as public, does that mean the proving key can be reused by other provers for other proofs?

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