The Death Note owned by Jack Neylon (his real name being Kal Snyder) was the one Light Yagami picked up in the first episode of the Death Note Anime Series, the one with some rules written on the inner side(s) of its cover. This is the very Death Note Light had in his hand when Kyosuke Higuchi (the person from Yotsuba) died of a heart attack. Since Higuchi was its previous owner, the ownership passed back to Light. Also, the Shinigami in possession of the Death Note before Higuchi was Rem.
So when, Light knew that Soichiro Yagami (his father) gave the Death Note to someone from Mello's gang (who they called Y462), he gave up the ownership of that Death Note. So the ownership now passed to Y462. Since this Y462 guy was eventually killed by the leader of the gang, the ownership must have passed to the next person who touched the Death Note, that logically being Kal Snyder. It is never mentioned that Kal Snyder never touched the Death Note. He must have. That is why he became its owner.
There are two logical ways in which the ownership of a Death Note can be affected after the 'death' of the Shinigami who owned it.
Let's suppose that if the Shinigami who owned a Death Note 'dies', the Shinigami who owned the Death Note previously, again becomes it's owner. In this case Ryuk becomes the owner of the Death Note, not Sidoh, as it was Ryuk who owned the Death Note before Rem. So in this case, Kal Snyder can see Ryuk, not Sidoh.
Let's suppose that if the Shinigami who owned a Death Note 'dies', the Death Note is no longer owned by a Shinigami. So, there is no Shinigami associated with the Death Note from the point when Rem 'died' to the point when Sidoh snatched the Death Note away from Mello. So in this case, Kal Snyder could only see Sidoh when he snatched away the Death Note from Mello and threw it at him. Not at any point before that.
By considering either of the above two cases to be true, we conclude that though the Death Note originally belonged to Sidoh, he was not currently it's owner. That is why Kal Snyder, or for that matter, anyone in contact with the Death Note (such as Mello) couldn't see Sidoh before the Death Note was thrown at Kal Snyder and touched by the respective person.