Have you tried to match permissions?
By default, Mac OS X formats volumes in journaled HFS+ volumes. Journaling is a feature that improves data reliability, and unfortunately it makes HFS drives read-only in Linux.
To disable journaling, just boot into OS X and fire up Disk Utility. Click on your HFS partition, hold the Option key, and click File in the menu bar. A new option to Disable Journaling will come up in the menu. Click that, and reboot into Linux. You should have read and write access to your HFS partition—however, the permissions on your Mac user's home folder will prevent you from reading or writing those files. we just need to change our UID in one OS so that it matches the UID in the other. Unless you have a reason for choosing otherwise, we're going to change our Linux UID to match our OS X one, since it's a bit easier. By default, the first user in OS X has a UID of 501, but you can double check this by going into System Preferences in OS X, right-clicking on your user, and hitting Advanced Options. If your User ID is something different from 501, replace 501 with your other UID in the terminal commands below.
Boot into Linux (we're using Ubuntu in this example) and fire up the Terminal. First, we're going to add a temporary user, since we don't want to edit a user that we're currently logged into. So, run the following commands in the Terminal, hitting Enter after each one:
sudo useradd -d /home/tempuser -m -s /bin/bash -G admin tempuser
sudo passwd tempuser
Type in a new password for the temporary user when prompted. Reboot and log in as tempuser. Then, open up the Terminal and type in the following commands, once again hitting enter after each one (and replacing yourusername with your Linux user's username):
sudo usermod --uid 501 yourusername
sudo chown -R 501:yourusername /home/yourusername
This will change your Linux user's UID to 501 and fix your home folder permissions so that you still own them. Now, you should be able to read and write to both your Mac and Linux user's home folder, no matter what OS you're logged into.
You may also want to fix your login screen, since by default Ubuntu won't list users with a UID of less than 1000. To do this, just open a Terminal and run gksudo gedit /etc/login.defs and search for UID_MIN in the text file. Change that value from 1000 to 501, and when you reboot your user will be listed in the login screen.