If you are a data processor it means that you are processing personal data on behalf of a data controller and it is the data controller not the processor which is in charge of establishing a lawful basis for processing personal.
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Consent forms and Privacy Notices are different things. The consent form is to be used to get the consent of the individuals for certain processing activities such as direct marketing, privacy notices on the other hand are meant to inform the individuals about the a certain processing activity.
Another difference is that privacy notices are just to be presented to the data subjects the data subject does not need to agreed to them.