In this scope, only answer you can get is a gathering of assumptions. So I will list them with explanations. Because there are no canon-wise information I can mention here that Dumbledore directly used.
In the Voldemort-Snape-Dumbledore triangle, there are so many layers you have to consider.
Snape's loyalties were always with Dumbledore, of course, after Lily's death. Snape was to stay undercover as a Death Eater, provide Dumbledore with inside information, and protect Harry at the same time. This was something Snape was ideally suited for: he was such an accomplished Occlumens that even Voldemort could not see his thoughts. As Dumbledore later told Snape, “To give Voldemort what appears to be valuable information while withholding the essentials is a job I would entrust to nobody but you.”
This means Dumbledore knows and trusts that Snape will give information to Voldemort sometimes by his choice of deciding or sometimes by consensus.
For example, when they used polyjuice potion to move Harry out they got caught because:
Snape: "My Lord, the Order intends to move Harry Potter from his
current place of safety on Saturday next, at nightfall."
Voldemort: "Good. Very good. And this information comes - "
Snape: "From the source we discussed."
He was talking about Mundungus. Snape confunded him and make him suggest the decoys to the Order. It was Snape's idea in the first place.
Even if this was after Dumbledore's death and Snape used Dumbledore's portrait to discuss plans sometimes, this means Snape made plans on his own.
You see, these kind of walk-arounds were Snape's area of expertise. And when you consider Dumbledore and Snape discussing things together, it makes sense if Snape gets creative with his feed of information to Voldy.
Since Dumbledore is the only wizard Voldemort fears, Dumbledore must have decided to give Voldemort vital information about himself through Snape. This way, he made Snape's seat among the Death Eaters and he was not afraid of Voldemort as much as others.
Although I don't think he mentioned Dumbledore's lethal error (of putting the horcrux-ring on) to anyone who was related with Voldemort, in order to keep horcrux information out of Voldemort's mind. This is why he only mentioned it like that in the Half-Blood Prince:
"Dumbledore is growing old. The duel with the Dark Lord last month
He had to assure Death Eaters that he was truly Voldemort's man.
Bellatrix: "But you didn't return when he came back, you didn't fly back
to him at once when you felt the Dark Mark burn —"
Snape: "Correct. I returned two hours later. I returned on Dumbledore's
Bellatrix: "On Dumbledore's — ?"
Snape: "Think! By waiting two hours, just two hours, I ensured that I
could remain at Hogwarts as a spy! By allowing Dumbledore to think that I
was only returning to the Dark Lord's side because I was ordered to, I
have been able to pass information on Dumbledore and the Order of the
Phoenix ever since!
Consider, Bellatrix: The Dark Mark had been
growing stronger for months. I knew he must be about to return, all
the Death Eaters knew! I had plenty of time to think about what I
wanted to do, to plan my next move, to escape like Karkaroff, didn't
I? "The Dark Lord's initial displeasure at my lateness vanished
entirely, 1 assure you, when I explained that 1 remained faithful,
although Dumbledore thought I was his man. Yes, the Dark Lord thought
that I had left him forever, but he was wrong."
Bellatrix: "But what use have you been? What useful information have we had from you?"
Snape: "My information has been conveyed directly to the Dark Lord."
He couldn't know Bellatrix and Narcissa were coming to his house that day in the Half-Blood Prince. So he improvized sometimes.
Also another point, Peter Pettigrew was staying with Snape.
"So, what can I do for you?" Snape asked, settling himself in the
armchair opposite the two sisters.
"We... we are alone, aren't we?" Narcissa asked quietly. "Yes, of course.
Well, Wormtail's here, but we're not counting vermin, are we?" He pointed his wand at the wall of books behind him and with a bang, a hidden door flew open, revealing a narrow staircase upon which a small man stood frozen.
"Wormtail will get us drinks, if you'd like them," said Snape. "And then he
will return to his bedroom." Wormtail winced as though Snape had thrown
something at him.
"I am not your servant!" he squeaked, avoiding Snape's eye. "Really? I was under the impression that the Dark Lord placed you here to assist me."
"I had no idea, Wormtail, that you were craving more dangerous assignments,"
said Snape silkily.
"This can be easily arranged: I shall speak to the Dark Lord--"
"I can speak to him myself if I want to!"
"Of course you can," said Snape, sneering.
Wormtail was placed there by Voldemort. This can also mean he was a spy for counter intelligence because Wormtail was listening to them behind a door, in secret. Wormtail is a coward character by creation he wouldn't listen 3 powerful (than himself) sorcerers just to please himself. He was acting on orders.
This can also mean that Snape was aware of the situation and intentionally letting out information so Wormtail can carry some valuable info to Voldemort as well. Maybe he was also telling these to Voldemort, in order to stay in Voldemort's good side (since he couldn't know which side Wormtail was... this can go forever because of the multi-layered espionage possibilities).
As a result, Dumbledore trusted Snape would do well by himself because he knew after a while Snape would have to go on this journey alone. They didn't know how many years this would go on. And decided that Snape would take charge at some points.