The pronunciation you describe is usually written "innit", and is heard most often in south-east England, especially London (but also elsewhere). In these dialects, the "t" in many positions becomes a so-called glottal stop instead of a real "t". A glottal stop is pronounced -- if that's the right word -- at the back of the throat (the "glottis") and is really when the air is stopped for a split second, which is why it's called a "glottal stop". Ask a Cockney to pronounce the word "glottal", and he'll probably say "glo'al".
So "innit" ends with a glottal stop, but in that position it's almost inaudible, which is why you hear "inny".
You don't have to pronounce it that way yourself if you can't: everyone will understand you if you say "isn't it".
And innit is a question tag. It is , however, becoming a very useful question tag, and you must have heard it where one would normally expect 'isn't he', 'aren't they', 'don't you' and several others.
Is it considered "correct" to use it this way
(saying 'isn't it' instead of 'isn't he'/'aren't they')
or is it merely "acceptable" given the propensity
of the masses to use 'isn't it'? Langauge,
being a living enity, changes. When I was younger, it was drilled into us that such usage
I love saying that there are no bad words in English, just bad situations. If you use 'innit?' instead of 'didn't they?' in a job interview, then it is definitely wrong, and it may mean you've wasted the money gettign that suit dry-cleaned.
It's DEFINITELY not correct usage in written English.
If you're just trying go about your daily business in London, you have to live with 'innit' on the bus, in the shops and in the pub.