That's the twist, the spren they're bonded to are in the Shardblades. If you think about it, we know that Szeth should be bonded to a spren but haven't seen it. We also know he has a Shardblade. How'd he get it? Someone made it using his honorspren! I can see two possibilities. 1) This is all part of being Truthless, the Shin can make the Blades, but they keep it very much a secret. 2) Szeth made it himself, which is why he's Truthless in the first place. The Shin weren't able to take it from him, though, or don't want it because of how it came to be.
Secondary theory, this sort of stems from #1 above. In one of the part 3 epigraphs (Jasnah's notes) it says they wanted to have Urithiru, the Radiants' HQ, in Alethela, but decided to move it farther west, "closer to Honor", I think it said. This could maybe put it in Shinovar, and that's how the Shin know how to make Shards. This could also support #2 above by having Szeth discover Urithiru and the knowledge hidden there to make a Blade. This is based on the assumption that the Knights knew how to make their own Shardblades. But it makes sense that Urithiru would be in Shinovar when you look at the Prologue and see how much he knows about Lashings and Stormlight and how it's connected to his religion.
Long, spur of the moment theories are my specialty.