Only in trivial cases where the training data isn't being curated properly. There was a paper done on the subject a few years back where "model collapse" was demonstrated by repeatedly training generation after generation of models on the output of previous generations, and sure enough, the results were bad. This result gets paraded around every once in a while to "prove" that AI is doomed. However, in the real world this is not remotely close to how AI is actually trained. You can prevent model collapse simply by enriching the training data with good data - stuff that is already archived, that can't be "contaminated."

Indeed, the best models these days are trained largely on synthetic data - data that's been pre-processed by other AIs to turn it into stuff that makes for better training material. For example a textbook could be processed by an LLM to turn it into a conversation about the information in the textbook, with questions and answers, and the result is training data an AI that's better at understanding and talking about the content than if it was just fed the raw text.

If so are these programs that claim to 'poison' the training datasets effective?

This is a separate issue from the usual "model collapse" argument. I assume you're talking about stuff like Nightshade, which claim to put false patterns into images that cause AIs to miscategorize them. These techniques are also something that only works in a "toy" environment, these adversarial patterns are tailored to affect specific AIs and won't work on other AIs they weren't specifically designed for. So for example you might "poison" an image so that a classifier based on Dall-E would become confused by it, but a GPT-Image classifier wouldn't care. The most obvious illustration of this is the fact that humans are a separate lineage of image classifier and these "poisonings" have no effect on us.

There's also the added problem that these adversarial patterns tend to be fragile, they break if you resample the image to resize or crop it. Since that's usually a routine part of preparing training data for an image AI it may end up making the poison ineffective even for image AIs that it was designed for.

Essentially, all these things are just added background noise of the sort that AI training operations already have mechanisms for dealing with. But they make people feel better, I suppose.