This is a slight variation of @gavenkoa's approach of using shared groups, and does not involve modification of the mounted drive.
Check what group id (GID) is used in the mounted drive.
cd /mnt/data/dir # or wherever it is mounted
Let us say you see something similar to the following:
drwxr-xr-x 3 12588 12000 4096 30.06.2017 11:22 Documents/
12000 is the current GID of the data in the drive, and the data can be read, and executed (cannot be written, though) by any other user of the group. Create a new group (for eg.
driveusers) and add your current user to the new group:
groupadd -g 12000 driveusers
sudo gpasswd -a $USER driveusers # add the current user to the new group
sudo chmod 770 /mnt/data/dir # optional, just to ensure the drive mount is accessible to user and group of the drive