Welcome to Android Central! I think that when you use Ringdroid to create a new ringtone out of an mp3 file, it saves that ringtone to the system's protected ringtone directory (not the user-accessible /Ringtones directory). I have a ringtone that I created from the Buckaroo Banzai theme, and it's not in the /Ringtones directory (although it shows up when I go to select a new ringtone using the Phone app's settings).
Here's a workaround on Ringdroid:
1. Open Ringdroid, find the ringtone you created, and select it.
2. On the next screen, tap the disk icon at the top, and select "Music" as the type. Then tap Save.
3. This will save the ringtone as an accessible mp3 file in the directory /media/audio/music. You should be able to share that file.