Andy Wilman, the show's producer who previously worked on Top Gear, said the team would instead judge the show's popularity from tweets.
"We’ll never know the number because Amazon won’t give us the viewing figure," he told the BBC. "You can just make something and never have to be judged by looking for overnights and consolidated figures."
Asked how he would know whether viewers had watched or liked the show, he said he would "get it from Twitter". He added that not knowing the viewing figures was "quite liberating".
When Top Gear was broadcast on Sundays on the BBC, Wilman and his team would conventionally have received overnight viewing figures on Monday.
But Wilman questioned the importance of overnight ratings, saying they were "dying".
"Over time, hopefully millions will watch, but we will say goodbye to that big figure coming in on a Monday morning," he said.
One reason The Grand Show is unlikely to replicate Top Gear's success, at least immediately, is because it is only available to Amazon Prime members. A subscription is currently available on offer for £59 a year ahead of the programme's launch, down from £79 a year.
The programme launches in weekly instalments from Friday 18 November, and is the first show from presenters Jeremy Clarkson, Richard Hammond and James May after the trio left the BBC.
Wilman said each episode's set-up was "expensive", but rubbished a reported figure of £4m per episode. But he admitted that each episode did cost more than £1m to make.
He added that, like the BBC, Amazon had left the team editorially independent and that it had made "nice noises" about the show.