To fix this, one of you needs to migrate to a separate account with a separate ID. You can still share the same ID for iTunes and other services; they don't have to use the same ID as you use for iCloud.
To migrate to a new account, start by saving any photo stream photos that you want to keep to your camera roll (unless already there) by opening your my photo stream album, tapping Select, tapping the photos, tap the share icon (box with upward facing arrow), then tapping Save to Camera Roll. If you are syncing notes with iCloud that you want to keep, you'll need to open each of your notes and email them to yourself so you can later copy and paste the text into new notes created in your new account. Then go to Settings>iCloud, tap Delete Account (which only deletes it from this device, not from iCloud), provide the password to turn off Find My iDevice and choose Keep on My iDevice when prompted . Then sign back in with a different Apple ID to create your new account and choose Merge to upload your data.
Once you are on separate accounts, you can each go to icloud.com from your computer and delete the other person's data from your accounts.