As root with file manager goto /media/nameofusb
right mouse click properties and change to watever you want
plus apply to all folders and submaps.
If you choose read and write for all users and groups you can use it on all Linux machines.
Your users have all a number, this is not the same with different distro's.
Thats why you cannot write to the usb, the maps/files are from a different user(number)
Or you have only read rights on NTFS (if not installed, ntfs-config in Synaptic)
A fat formatted usb has no rights management, but has large file problems.