Keeping your email on your devices, and off of third-party servers, means anyone who wants to look at your email has to come to you and not your email provider.
There are a few weaknesses to that argument, however, since intelligence agencies could still grab your email while it transits the Internet. Your mail provider may also have redundant backup copies of your email that won't get deleted right away, defeating the whole point of using POP3 for privacy concerns. (Fortunately, it's possible to encrypt your email.)
Sure, most of us are probably never going to be targets of a police investigation. But for most people these privacy concerns are really about the principle of the issue .
The downside of POP3 in a multi-device world is that you'll have to take some precautions and think hard about how to access email on a mobile device.
Since the sole copy of your email is now on your PC, you'll have to have a solid back-up plan to make sure you don't lose your messages to a failed hard drive.
As for smartphones and tablets, you should still use IMAP there if possible, even if you're using POP3 on your PC. The last thing you want to do is download email to both your phone and your PC via POP3, since you'll end up with two separate repositories of email: stuff downloaded to your phone and stuff downloaded to your PC. It's a nightmare.
If you're using a major webmail service like Gmail, the easiest thing to do is just use Google's Gmail app for Android or iOS. Ditto for Outlook.com and Yahoo Mail.
One last note about IMAP on your phone and POP3 on your PC: if you reply to email on your phone, your PC won't download new messages in your sent folder, since POP3 only grabs messages from the server. So for POP3 users, mobile devices are better for viewing or deleting email, but not necessarily for sending messages you may need a paper trail for later.
Also, remember that if you leave your mail client running on your PC while you're out, all mail messages could disappear from your phone as your desktop grabs new batches of email--unless you (usually manually) configure your email to continue to store messages on the server for a predetermined length of time after you download them.
It's not a perfect solution, especially if you need mobile access, but if storage quotas or privacy concerns are issues for you, then POP3 is probably a better choice than IMAP.
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