That folder is not Dropbox as in Dropbox (the file sharing software) , it's just a Dropbox as in Drop Box, (a place to drop files to be received by that user).
Apologies if your brain just exploded trying to read that, but it's accurate. You could think about is Like a letterbox in a door.
Lets explain how that works.. To do so let's have a look at my system (in the transcript below I've replaced x64's super-secret real name with 'x64' - no electrons were harmed in that exercise!, I've also added a splash of colour - red is my typing, and blue the shell's prompt and some labels I've added to highlight some lines of interest)
MBP:~ x64$ ls -l /users
drwxr-xr-x+ 12 Guest _guest 408 5 May 10:03 Guest (*** line 1 ***)
drwxrwxrwt 7 root wheel 238 8 May 20:57 Shared
drwxr-xr-x+ 21 x64 staff 714 25 Jul 19:06 x64
MBP:~ x64$ ls -l /users/Guest/
drwx------+ 3 Guest _guest 102 5 May 10:03 Desktop. (*** Start of lines 2 ***)
drwx------+ 3 Guest _guest 102 5 May 10:03 Documents
drwx------+ 3 Guest _guest 102 5 May 10:03 Downloads
drwx------+ 26 Guest _guest 884 5 May 10:03 Library
drwx------+ 3 Guest _guest 102 5 May 10:03 Movies
drwx------+ 3 Guest _guest 102 5 May 10:03 Music
drwx------+ 3 Guest _guest 102 5 May 10:03 Pictures (*** end of lines 2 ***)
drwxr-xr-x+ 4 Guest _guest 136 5 May 10:03 Public (*** line 3 ***)
MBP:~ x64$ ls -l /users/Guest/Public/
drwx-wx-wx+ 3 Guest _guest 102 5 May 10:03 Drop Box (*** Line 4 ***)
So what does that tell us.... firstly consider 'line 1' - the permissions for the Guest folder itself start with 'rwx'... Guest has full access to its own home folder.. Seems fair to me... Then ...'r-xr-x'... other users in the same group as Guest and everyone else in general has read address (huh? - doesn't seem very private for poor old guest, does it... but hang on....)
Consider 'lines 2' - the folders within /users/Guest. Again the permissions start with 'rwx' so guest has full access to it's stuff, but then the permissions continue '------' no other group or other users have any access to those folders (more like what we's expect of user segregation).
Then there is 'line 3'. A public folder alongside the private folders of the Guest user - others still have Read access to that - OK - Amongst other uses, Guest could put stuff there to be shared with other users.
And finally 'line 4' The drop box folder... Again Guest has 'rwx' access - everything, other members of guest's group and other users in general have '-wx' access - effectively write access (specifically without read access... ). So Otheruser1 could write a file there and Otheruser2 could do the same, but Otheruser1 could not see the file that Otheruser2 dropped in that folder (and vice versa). Guest however could see both files. OK - that's the privacy thing accounted for - '~/Public' and '~/Public/Drop Box' is a file exchange mechanism....
Re the dates - I received my MacBook on 5th May which explains that date for my folders. any other similar notable reason for 14th may being significant on your Mac?
I'd not go around deleting these folders. I can't see them doing any harm (as long as the Guest account is disabled, and their non presence might cause other issues for software or updates that expect to need/touch them)
I've used Guest as the example, but the same mechanism exists for sharing files between any two users.
Edited by x64, 25 July 2017 - 03:13 PM.