There's a nice explanation of the behaviours of squirrels at LiveScience.com. Basically it boils down to that they can remember about 95% of the hoards, but they will use deception on the burying location when other squirrels are around. This suggests that squirrels do indeed intentionally dig up other squirrel's nuts.
As to Jays - Jays are well known to have excellent memory and can find almost all of their cached food. They have also been observed stealing cached food from other jays and other animals (see the "cool facts" section).
So - the answer is yes, both will steal and both remember where their caches are.
As an aside: If you have ever read the story "The Tale of Timmy Tiptoes" by the beloved English author and illustrator Beatrix Potter to any children in your life, then you may be familiar with the phrase "Who's been digging up my nuts" kicking off the central conflict between squirrels in the story. Potter was an astute observer of animal nature and a talented nature artist, who would have observed the native Red Squirrels, which also exhibit caching behaviour. The protagonist (Timmy Tiptoes) in the book is an Eastern Grey Squirrel as the book was written for the American market, which she viewed at the London Zoo for her illustrations..