You probably use your own spell save DC, but RAW is unclear
The description of the Ring of Shooting Stars (DMG, p. 192) does indeed not give a spell save DC for faerie fire. It also gives access to dancing lights and light, the latter of which may also require a spell save DC ("if you target an object held or worn by a hostile creature"), which the item's description does not provide either.
Generally, when a magic item provides a DC for multiple abilities or spells, it's always the same DC for each ability/spell, so a DM may well adjudicate that the spell save DC for these spells should be 15, the same as the Ball Lightning or Shooting Stars abilities.
However, although this magic item doesn't require attunement from a spellcaster, it may well be the case that you are meant to use your own spell save DC, as suggested in the DMG (p. 141):
A magic item, such as certain staffs, may require you to use your own spellcasting ability when you cast a spell from the item.
This could be read to mean either:
that the default assumption is that you use your spellcasting ability unless it gives a specific DC, as per the web spell from the Cloak of Arachnida (DMG, p. 158):
- You can use an action to cast the web spell (save DC 13). [...]
or that it is not the default assumption and that a magic item must explicitly state that it uses your spellcasting ability, as per the Staff of Fire (DMG, p. 201):
[...] cast one of the following spells from it, using your spell save DC: [...]
Since it is not clear which way this is meant to be interpreted, the DM may have to adjudicate whether or not the item should use your spellcasting ability (since it doesn't mention a specific DC) or a specific DC (since it doesn't explicitly mention that you use your spell save DC; see above for reasoning around why this specific DC would likely be 15).
Of these two options, it makes sense to me that using your spell save DC is the more likely conclusion, since I think explicitly mentioning your spell save DC is less specific that giving a specific DC, but really any DM can pick their preferred interpretation, since RAW is not clear at all.
Furthermore, it seems that D&DBeyond also chooses this interpretation (thanks to Sdjz for pointing this out), since (as you can see with this test character I created) it uses my character's spellcasting ability modifier (which you can see by going to the Spells tab and seeing the HIT/DC column for both light and faerie fire; note that I deliberately made a barbarian to show the effect of the below quote).
For classes that don't have spell save DCs (because they are not spellcasters), these two Q&As suggest that, in this case, you would assume a DC of 8 + 0 + proficiency, as per the DMG (p. 141):
If you don't have a spellcasting ability - perhaps you're a rogue with the Use Magic Device feature - your spellcasting ability modifier is +0 for the item, and your proficiency bonus does apply.