EU is more complicated, but Article 14.5.b allows withholding notice if it would impair/defeat the purpose of processing. The PI must however apply "safeguards", whatever it could mean.
Pretty sure that would be considered stalking and is broadly illegal in the US, PIs being an exception.
This becomes extremely relevant when you read it in the light of the C-422/24 decision. In that personal data collected via body worn cameras was determined to be "directly obtained". Paragraph 41 from the judgement:
> If it were accepted that Article 14 of the GDPR applies where personal data are collected by means of a body camera, the data subject would not receive any information at the time of collection, even though he or she is the source of those data, which would allow the controller not to provide information to that data subject immediately. Therefore, such an interpretation would carry the risk of the collection of personal data escaping the knowledge of the data subject and giving rise to hidden surveillance practices. Such a consequence would be incompatible with the objective, referred to in the preceding paragraph, of ensuring a high level of protection of the fundamental rights and freedoms of natural persons.
Given this it's very unlikely that PI observing (especially if they record) could be considered to be Article 14 instead of Article 13 type of collection as it's exactly "hidden surveillance practice" that the Court warned about.
Member states do have a right to restrict the Article 13 disclosure obligations via Article 23 restriction, but that requires specific law in the member state & the law itself must fulfill the obligations that Article 23 requires. Article 23(2) essentially forbids leaving everything up to the controller.
And as far as PI in the US goes, actions between stalking and PI "for self" tend to be so similar that I wouldn't necessarily recommend anyone to try it.