There is no security issue when making getter functions, period!
A getter function is constant (typically view), hence does not change the state.
Therefore, such function cannot yield a security breach by definition.
You may have a security issue in some non-constant function which calls some getter function, but that security issue will not be resolved ...
Provided you are using the same algorithm that ETH uses for your private chain, you will always have to mine in order to create blocks and process transactions.
Since it is a private blockchain, there should not be many miners on the network, and thus the hashing power should be low. Because of this, you are able to mine with a simple CPU and very little ...
Signing a transaction is a one way cryptographic operation. The private key cannot be derived from the signature. Moreover, the signature must be readable because it verifies you are the creator of the transaction. Also, all transactions are public. So, yes your wallet is safe.