Hi @ylor,
Not necessarily so. I typically separate out my different amplicons / data sets prior to denoising. IMO, as long as you have enough data for the denoiser to make reasonable estimates, you should be fine. I doubt any differences would be significant. Also, many users perform additional taxonomic and sequence based quality control in addition to denoising anyway.
Just run cutadapt trim-paired separately for each primer pair with the --p-discard-untrimmed option set. This flag will force cutadapt to only write out the paired-end reads to file in which the primers were detected and trimmed. This way you have have a file for each primer set, and you can keep track of everything that way.