Can you cast a spell with split second after your opponent casts a spell with split second?

Split second (CR 702.61) means that as long as that spell is on the stack, players cannot cast other spells or activate abilities that aren't mana abilities. This lock applies to all players equally — including the player who did not cast the split-second spell.

Crucially, split second does not create a layer of priority exceptions. It is a continuous effect that simply prohibits casting spells while the split-second spell is unresolved. Because your opponent's split-second spell is on the stack, you cannot cast anything — including another split-second spell — until that spell resolves, is countered, or otherwise leaves the stack (CR 702.61b).

The only actions still permitted while split second is active are: playing lands, activating mana abilities (CR 702.61b), and special actions that don't use the stack (CR 116.2). Triggered abilities can still trigger and go on the stack as well, since triggers aren't "cast" or "activated" in the relevant sense.

Concrete example: Your opponent casts Sudden Shock (which has split second) targeting your creature. You want to respond by casting Krosan Grip (also split second) to destroy a problematic artifact. You cannot — Sudden Shock's split second locks out all spell casting until it resolves, so Krosan Grip must wait until after Sudden Shock has resolved.

The net result is that split-second spells cannot "race" each other once one is on the stack. Whichever split-second spell hits the stack first resolves without any other spells being cast in response.