From what I understand of the not-very-detailed text watermark, it comes up to masking the generated output with specific bias in the weights of the token generation (or something like this). Anthropic was explaining that it would survive edits (not full rewrite). So asking another AI to paraphrase will surely add its own, but since the bias is not disclosed, it will depend on how the 2nd AI is considering the existing tokens to steer its own.

Fictive example: imagine the bias is to inject notion of colorfullness in the text regardless or its content:

"I eat an apple" becomes "I eat a red apple"

The 2nd AI is about injecting size component, so the text becomes " I eat a big red apple"

There you get both watermarks. It is not that obvious obviously