The terms of service for Open AI assign to the user all its rights, title and interest in and to 'content', where 'content' includes all inputs and outputs.
Google is less clear in its assignment of copyright, but it does provide that the user's content includes anything the user creates, uploads, submits, stores, sends, receives or shares using Google's services.
The question of whether the user is creating an output or whether Google Bard is 'creating' an output is left open. The IPO's report on its consultation implies that in the UK, the creator of the output would be seen to be the AI, and so following Google's Terms and Conditions, ownership would revert to Google.
To conclude this discussion, let's look at this from a practical perspective. Given the nature of the offering by Google and Open AI and their business models, it is very unlikely that either provider would challenge the use of content generated by their respective AI technologies (unless the user is in violation of the relevant Terms of Service). Therefore, in practice, it wouldn't really matter who the owner would be.
Irrespective of who is understood to be the owner of the copyright, the dilemma persists with regard to the requirement for 'originality', especially where the creator of the content (the AI) and the recognised author (either the user or the service provider) are different.
Interestingly, the US Copyright Office has issued a policy statement providing guidance on whether work created with AI is eligible for copyright protection.
The Copyright Office stated that the determining factor for originality would lie in whether the AI contribution is a result of 'mechanical reproduction' or a user's 'original mental conception'.