At the start of the Avengers: Age of Ultron movie, the Avengers are on a mission, attempting to retrieve Loki's staff from HYDRA. They're attacking a castle which is a HYDRA base run by Strucker and they've got a Chitauri Leviathan corpse at the base.
Presumably this came from the battle of New York from the first Avengers movie, but how did HYDRA get the corpse of a Leviathan out of New York without S.H.I.E.L.D doing anything about it?
As noted in the comments, as seen during the events of Captain America, Winter Soldier, Hydra had infiltrated every level of SHIELD by the 2010s. We know that SHIELD was already doing alien weapon research (The Destroyer weapon that Coulson used) and Hydra Tesseract weapon research (Hydra guns Cap found on the helicarrier). They even "converted" Hydra scientists, like Arnim Zola, whom in turn decided to poison SHIELD from the inside.
By 2010s, Hydra agents were everywhere, from low level rank and file, special agents (Jasper Sitwell), elite soldiers (STRIKE team leader Brock Rumlow), the World Security Council (Alexander Pierce), and even senators (Senator Stern, the prick from Iron Man).
These two factors combined, would have meant SHIELD either intentionally set up to study the Leviathan for further benefit, or Hydra intercepted it, as they did the Scepter in the Winter Soldier stinger.
It's highly likely that the Hydra base was a black site bankrolled by SHIELD until it collapsed.
By using SHIELD.
By this point, it has been revealed (in Captain America: The Winter Soldier) that SHIELD had long ago been compromised by Hydra, the organisation that Baron Von Strucker was an important part of.
As a high ranking member of Hydra, he would have been able to have his people within SHIELD deliver the Leviathan corpse to him when the clean up operations were taking place following Battle of New York (seen in Avengers Assemble). We know that SHIELD were an important part of the clean up operation thanks to the short film Marvel One-Shot "Item 47", which sees Agents Sitwell and Blake attempting to tie-up some lose ends from the Chitauri Invasion.
