Pretty sure this is, at least in my case, closely related to having multiple users logged-in, using fast user switching (where the desktop rotates on a cube when switching users). This "pops up and disappears" thing always seems to happen when I have two accounts logged-in.
Of course it always gets fixed with a reboot, but you shouldn't have to reboot to use a USB device, just because the "allow/deny" box, which should be front and center on the operational screen, popped up and then decided to hide itself in desktop compositor limbo.
If you've got multiple accounts logged in... does logging one out help? (Apple menu, "Log out Name...")
If you're not sure, check Activity Monitor and, under the CPU tab, scroll the table to the right side, and click the "User" column to sort. Then, check to see if there's another username besides yours, service accounts (starts with an underscore _), and "root" users. If there's another name (with lots of processes), there's another account active. If there's another name but it's only one or two, well... that's odd? Could be a clue.
If there's no other accounts logged in, then we're back to square one.
Wish it were possible to report issues like this to Apple, but it's definitely a bug, and quite an annoying one in certain situations.
