I may be wrong, but with IMAP, I thought the client only had the *headers* from the mailbox, unless the user chose to (a) download the body of the message for viewing, or (b) copied from the IMAP mailbox to local folders (and thus caused the body to be downloaded). That's part of the beauty of IMAP, you don't have to download GB of spam (imagine various 5+ MB HTML brochures with loads of images in), you can decide from the headers what you wish to view in full, and then just download the body of those messages you choose to view.
Means that from a mobile, one can find the message about a meeting being cancelled, or travel / hotel arrangements, without having to download lots of other messages, which could incur significant charges (eg if roaming while abroad).
These days, I'd recommend people get separate web/mail hosting than depend on ISP-based mail. For one thing, it means switching ISP is far easier, and for another, it can be assumed that the priorirty for the service provider is to handle/manage that hosting and keep it secure. Many ISPs offered "extras" but one by one those have gone away. For example, years ago, it was common to find an IRC server, and USENET news, but those are seemingly rare (at least in the case of NNTP with binaries).
If you run a business, have a second ISP and backup web hosting...