(Not sure what the little pyritic explosions at either end are, any suggestions welcome.)
But if you're rummaging among the shingle for fossils you are likely to see this kind of thing:
They're cylindrical bits of smashed up belemnite rostrum, the pointed specimen is a whole (but heavily abraded by being tumbled in shingle) rostrum. The rostrum was basically the belemnites's tailbone to which its fins attached. What belemnites looked like and how they lived was a matter of debate between pelaeobiologists because most of their living bodies were soft tissue that is preserved in fossil form rarely and then usually poorly.
However, if you're curious to see what enclosed these fossil cylinders and bullets in the Jurassic seas of Yorkshire (and elsewhere) go and read Adaptations to squid-style high-speed swimming in Jurassic belemnitids from the January issue of the Royal Society's Biology Letters. The paper analyses specimens where belemnite soft tissue had been fossilised and concludes that they were fast-swimming creatures that lived in the water column, rather than grovelling around on the Jurassic seabed.
Update: And thanks to Brooke Johnson for this: Last Suppers: Calamari del Mesozoico. More excellent stuff about belemnite soft tissue preservation and lots of good stuff about food chains and predation in Jurassic seas, which is just what Runswick Bay was 190 million years ago.