How should influencers disclose that this is a paid partnership?

It is the influencers' responsibility to follow all guidelines set forth by the FTC or other governing bodies.

The U.S. Federal Trade Commission (FTC) has published guidelines on required disclosures, which include: 

FTC Disclosure examples

Source:  FTC's Disclosures 101 for Social Media Influencers

 

We recommend you familiarize yourself with these guidelines in full, however, you can trust that our contract and terms require the influencer to follow all applicable rules and laws. Our standard influencer agreement, which all influencers sign as part of their campaign contract, attests that the influencer will follow all disclosure requirements: 

Influencer Agreement - Disclosures

Read the full influencer agreement here.