My dog thought this was a great idea, just need some advise on what type finish is on the wood currently? I'm going to sand the chewed part to start.
Unless you want to refinish the whole table, this is really just going to come down to trial and error. Find some scrap wood, head down to the store to find some similar looking stains / oils / etc., and keep trying on scrap until you find something. You're probably not going to get the finish exactly right. You might also want to take a look through these links about distressing wood.
The damage is too extensive to fill it with putty, you'll probably want to remove that corner and replace it, or remove and replace that entire end piece.
Another thing you could try is changing the shape of the table a bit. E.g. maybe if you cut all corners at 45 degrees (to remove the chewed part and give the table a chamfered corner look), or cut a flat inch or two off the damaged side it could turn out well. But again you'll have to experiment with finishes, although you don't necessarily even have to match the finish... Another idea could be to cut the corners off then get creative, perhaps strip all the finish off just the sides and give it a different color, like a darker band around the edges or something, or even trim it with a different kind of wood. (As a bonus this'll also hide that splotch of gray stuff on the right in your picture.)
By the way, your dog will probably keep chewing it, you also might want to look into ways to change that behavior to prevent further damage in the future.
Warning: I don't actually know what I'm talking about and have limited experience.
Round all the corners (to hide the fact that just one was damaged), and then finish the newly exposed wood? As long as all the corners match each other, folks will assume this was a deliberate design decision, even if they don't match the rest of the bench.