In order for a user to be able to see the contract they are responsible signer for, they will need to be able to see the 'Files & Forms' section for the student they are 'responsible signer' of.

This means that the user will also need the 'parental access' flag on the same relationship that they have the 'responsible signer' flag for.

Responsible Signer and Parental Access Flags on the Relationship

If it is important that this user not have parental access, there is one alternative to allowing them to keep the Parental Access flag, but it is manually involved.

The base reason that the user does not see the contracts/forms, is that without Parental Access, they do not see the 'Files & Forms' for the student:
Files & Forms

However, if a user that does have access to the Files & Forms section, the URL for the contract will remain the same (as long as it is not completed or re-generated/re-issued)

Contract URL

This URL can then be shared with the user that does not have 'Parental Access', and they can access it once they are signed in.
This will allow them to complete a contract/form of a student they cannot otherwise see the Files & Forms section for.