It looks keratinous, almost like... Well, like the tomium (beak) of a turtle. Noting the size, it would have to be a sea turtle. It also looks somewhat vascularized. Without actually getting my hands on it, I would have to say a sea turtle.
You say the Mediterranean? Hmm... Green turtles Chelonia mydas nest in the Azores, and make their way into the Mediterranean a lot.
The link below has a photo of a Green turtle skull (a replica) with its beak attached. The mystery thingy looks a bit more narrow and pointed...do young Green turkles have more aggressive beak profiles than mature tutles?
Yes they do. Along with some other features that are designed to make them less palatable to larger animals. As far as the bird theory goes, my wife took a look at the beak thing and said that she didn't think it was a beak due to the fact that bird beaks are the result of a fusion of three bones, and this thing doesn't show signs of that process.
Could also be a tortoise as well, but just looks a bit big.
Are we sure they're aquatic or marine in origin? Lots of weird stuff washes down rivers, especially if they run through farmland.
My first thought on this is that it looks very hooflike; I did some checking around and the vascularization seen distally in many hooves (like the second one on this page at horseshoes.com) does look similar.
See attached (also courtesy of horseshoes.com); the 'toe' of the hoof (which may often become overgrown and clawlike if not trimmed or worn away through regular activity) has the right cavity shape when you remove the coffin bone / distal phalanx.