Several of the methods suggested will work for at least some stains.
More severe stains that produce etching probably can be completely
removed only by polishing the glass on the inside using shot and very
fine grinding and polishing compounds and a machne that rotates the
decanter for a long time. Unless you polish rocks as a hobby, this is
something for a professional, and would be worthwhile only for a very
expensive item.

In the US, there is a commercial product called CLR that often will
remove many calcum-lime-rust types of stains(hence the name). It can
clean up coffee pots that can become very stained. It is widely
available in some home improvement stores and supermarkets. Be sure to
read the instructions carefully. It contains glycolic,sulfamic, and
citric acids as well as surfactants. If it works, it works fairly fast
and long contact is not suggested for most surfaces. I have used it on
coffee pots and a rust stain on a bath tub, but not on a decanter yet.
This might be a good last resort if milder cleaning methods fail and you
can not justify the expense of professional polishing.

My mailbox is always full to avoid spam. To contact me, erase
from my email address. Then add . I do not
check this box every day, so post if you need a quick response.