A victim typically doesn't make saving throws against the functions ball lightning and spark shower
The description of the ring of shooting stars (Dungeon Master's Guide 232–3) (50,000 gp; 0 lbs.) says that victims make a saving throw against the special function shooting stars but does not say that victims make saving throws against the special functions ball lightning and spark shower. As the ball lightning and spark shower functions of the ring of shooting stars are not said to be like other spells, those functions are unique, operating the way the ring says they do therefore not offering their victims a saving throw.
While the Dungeon Master's Guide on Rings does say, in part, that a magic ring "has a spell-like power (often a constant effect that affects the wearer)" (211), in context, that's from an introduction to magic items and not from, for instance, Saving Throws Against Magic Item Powers that says, "Most item descriptions give saving throw DCs for various effects, particularly when the effect has no exact spell equivalent (making its level otherwise difficult to determine quickly)" (214). I urge not inferring from the basics of magic items how specific magic items function.
For comparison, a ring of invisibility (232) (20,000 gp; 0 lbs.) allows a wearer to take a standard action (see here) to "benefit from invisibility, as the spell." Understanding that benefit requires the reader to visit the 2nd-level Sor/Wiz spell invisibility [illus] (Player's Handbook 245) to learn that each activation of the ring's benefit has a duration of 3 min. (the ring's caster level; see here) rather than, for example, the ring making the wearer invisible while it's worn and visible again when it's removed. Further, the reader learns that only the wearer benefits from the ring—the wearer can't render invisible an unattended object nor an object another's attending, for instance, as can a normal invisibility spell (or, for that matter, spell-like ability, like that of an ogre mage (Monster Manual 200)).
However, unlike the ring of invisibility and the spell invisibility, no spells are referenced in the description of the special functions shooting stars, ball lightning, and spark shower of the ring of shooting stars. The reader is left to rely on the descriptions of those functions as presented in the ring of shooting stars description alone, and only the shooting stars special function of the ring of shooting stars mentions victims make saving throws—therefore victims of those other two functions don't.
Note: The spell ball lightning was originally published Aug. 2001 in Magic of Faerûn (80), after the ring of shooting stars appeared in the Sept. 2000 Dungeon Master's Guide. An argument can be made for Guide authors anticipating Magic therefore the ball lighting spell, but one goal of Third Edition was to reduce the consumer's need to reference other products. Including in the core rules a reference to an upcoming product—and to not even mention that upcoming product's title!—runs counter to this goal. Also, I am aware that the ball lightning function, in part, says that spheres "impact for 12 points of damage and spread (as a fireball [spell]) in a 5-foot-radius sphere" (232), but that reference to the spell fireball is likening only that spread to the spread of a fireball spell, and not likening the entire ball lightning effect to a fireball spell. (Were it doing the latter, the parenthetical would have to be at the sentence's end.) Finally, that reference is mechanically inaccurate—a fireball spell's point of origin is a grid intersection, and the sphere's 5-ft. radius extends from the creature struck… and whether that means it's actually a 2½-ft.-radius spread or actually a 7½-ft.-radius spread is anybody's guess. (Mine's the latter.) Finally, unless the ring of shooting stars has narrative value, this player recommends selling it as soon as possible—its price inadequately reflects its utility.