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Sam - Near Crith Ungol:

Sam puts on ring and " Sam hurled himself against the bolted brazen plates and fell senseless to the ground. He was out in the darkness"

He put the ring on before he goes unconscious. He woke up with the ring and The Dark Lord did not detect him.

Already the Ring tempted him, gnawing at his will and reason. Wild fantasies arose in his mind; and he saw Samwise the Strong, Hero of the Age, striding with a flaming sword across the darkened land, and armies flocking to his call as he marched to the overthrow of Barad-dûr. And then all the clouds rolled away, and the white sun shone, and at his command the vale of Gorgoroth became a garden of flowers and trees and brought forth fruit. He had only to put on the Ring and claim it for his own, and all this could be.

Frodo - At Mount Doom:

‘I have come,’ he said. ‘But I do not choose now to do what I came to do. I will not do this deed. The Ring is mine!’ And suddenly, as he set it on his finger, he vanished from Sam’s sight. Sam gasped, but he had no chance to cry out, for at that moment many things happened.

Something struck Sam violently in the back, his legs were knocked from under him and he was flung aside, striking his head against the stony floor, as a dark shape sprang over him. He lay still and for a moment all went black.

And far away, as Frodo put on the Ring and claimed it for his own, even in Sammath Naur the very heart of his realm, the Power in Barad-dûr was shaken, and the Tower trembled from its foundations to its proud and bitter crown. The Dark Lord was suddenly aware of him...

marked as duplicate by Jason Baker, The Fallen, Shevliaskovic, Meat Trademark, phantom42 Sep 5 '15 at 14:57

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No. But it helps.

When Frodo puts the ring on at Amon Hen Sauron is aware of him and is closing in on finding him.

Gandalf says in The White Rider

The Ring now has passed beyond my help, or the help of any of the Company that set out from Rivendell. Very nearly it was revealed to the Enemy, but it escaped. I had some part in that: for I sat in a high place, and I strove with the Dark Tower; and the Shadow passed.

And in the tower of Cirith Ungol Sam realises he can no longer use the ring now he has passed into Mordor

‘And anyway all these notions are only a trick,’ he said to himself. ‘He’d spot me and cow me, before I could so much as shout out. He’d spot me, pretty quick, if I put the Ring on now, in Mordor. Well, all I can say is: things look as hopeless as a frost in Spring. Just when being invisible would be really useful, I can’t use the Ring! And if ever I get any further, it’s going to be nothing but a drag and a burden every step. So what’s to be done?’

The reason Sauron misses Sam at Cirith Ungol is because he's not looking for him there, he assumes one of Gandalf/Aragorn has the ring in Gondor and is looking that way.

So just wearing the ring won't draw Saruon's immediate attention but it does make you visible to him.

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    Also the nazgul spotted Frodo clearly when he put the ring on at Weather top and at that point Frodo had no intention of trying to claim it,and as pointed out in the answer Sauron was aware of him at Amon hen,so the answer is whether you claim the ring or not won't make a huge difference ,simply using the ring can lead to Saurons detection of it – turinsbane Sep 16 '15 at 20:09

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