https://www.chess.com/legal/user-agreement
"While using the Chess.com service, your game activity will generate moves, games, ratings, and statistics. This data becomes publicly available and cannot be hidden or removed. Additionally, all chess moves and games are public record and can be freely accessed, copied, and used by anyone for any lawful purpose."
Seems to cover some of what you mention. But probably an email to support would get you a clearer answer.
https://www.chess.com/legal/user-agreement
"While using the Chess.com service, your game activity will generate moves, games, ratings, and statistics. This data becomes publicly available and cannot be hidden or removed. Additionally, all chess moves and games are public record and can be freely accessed, copied, and used by anyone for any lawful purpose."
Seems to cover some of what you mention. But probably an email to support would get you a clearer answer.
Thanks TSG. 
Haha. Yes I should have read the User Agreement - I'll look over it again.
One interpretation of the section you quote from the UA is that you can't get upset if someone uses your games and/or personal data elsewhere. It seems to suggest what I'm proposing is OK. But, I guess my query is slightly different as it is from the point of view of the person using the data, and I guess depends on what "lawful" means in this case.
I'm just wondering if there are any guidelines on the legality of reproducing or republishing chess.com data elsewhere. I am working on a few ideas for chess data-science projects and it'd be great to be able to publish datasets on other sites (e.g. Kaggle). The user agreement, as I'd expect, (and as far as I can tell) states that we're not allowed to "upload, post, email, transmit or otherwise make available" anything, but obviously the provision of the public APIs means that chess.com wants us to use data elsewhere for various purposes.
Obviously what I'm envisaging is not wholesale copying of game archives but could be derived datasets, like:
Here's one example of a dataset republished from data derived from the other site on kaggle: https://www.kaggle.com/datasets/mysarahmadbhat/online-chess-games. A dashboard-like example is http://chessleagues.com/. Copyright considerations of blog posts of data analysis from chess.com API data (rather than publishing the data itself) on data analysis of chess.com derived data is something else I've been thinking of too. Are there any issues with something like that?
Any guidance that dev community members or admins could provide would be helpful in what is legal and/or acceptable in relation to this would be very much appreciated.
Thanks.