@Matthias
First of all yes, as you imagined, only contacts that are saved on the phone can have a personal ringtone.
If your contact is on the sim card, then there won’t be any option to set the ringtone. 
Second, you need to make sure that the desired ringtone is indeed in the Ringtone directory.
Now the procedure:
- Launch the “People” application
- Select the contact (a new page/view will open)
- Tap on the “Settings” button (the one on the lower left, next to the home button) and a dialog will open
- Tap on “Set ringtone” as shown in the attached image
(edit, by Herve5)
I am quite ashamed to edit a post not mine (I don’t even understand why this is possible -maybe because the discussion is closed); this is to say, the procedure to move a contact from the SIMcard to the phone memory is not obvious, while most certainly the situation where all contacts are in the SIM will happen to most newcomers.
The only way I found is asking to ‘associate’ said contact with a given Sim (out of the two), which triggers the duplication of the contact into the phone memory. Then by editing that second contact one indeed has access to dedicated ringtones, contact avatar etc.
Sorry again for this shameless edit…
H.