They say that just to offer encouragement. Don't believe them.

First a quick note - normally when Chinese talk about "dialects", what they mean is "languages". For example, Shanghainese and Mandarin are two different Chinese languages, not two different dialects of Chinese. Not a big deal, but I thought our Chinese members might want to know how to correctly talk about them in English.

I don't find it encouraging when I get told how great my Chinese is after 你好, but this is common behavior of native speakers in all the languages I learn. As others have mentioned, tones are more important to Mandarin speakers than initials and finals. That's because there is a lot of acceptable regional variation of initials and finals, but little variation of tones. It's not "wrong" to pronounce "shi" as "si"; it's an accepted variation, common in parts of the South. It's not "standard Mandarin", but it's not wrong. However, pronouncing the 1st tone as a 4th tone, which many Chinese who don't grow up speaking Mandarin occasionally do incidentally, isn't acceptable in most cases.

That being said, I sometimes hear Taiwanese pronouncing words with different tones, usually using 1st instead of 4th, than northern mainland speakers. But this type of thing is much rarer than variations in initials and finals.

So if they think you sound like a 外國人, it's probably tones. But it might also be something that's rarely taught. There is often a stressed word in a sentence. "When the tone(s) in a stressed word is mispronounced, the sentence will frequently sound quite bad to native ears, but when the stressed word is pronounced correctly, the other tones will often fall in line." Here is an article about this:

http://www.sinosplice.com/life/archives/2008/12/10/toward-better-tones-in-natural-speech

I would say "Your Chinese is very good" if I had no difficulty understanding what a learner of Chinese said, because I don't think it matters whether they have a strong Italian/American/Russian accent.

It's true that many Chinese, especially most people from south China, don't speak Mandarin very well either, but unlike foreigners, they can pronounce most of Chinese words in the correct tone. This might be the main reason why these Chinese don't sound like foreigners at all when they speak Mandarin.

We won't ask a foreigner to speak Chinese as well as a native speaker, so if we can understand what they say, we will say that they speak will. As for the people from south China(I am from southeast), there is a different standard.  

You may wonder how can they tell the difference from foreigner and native speaker. I think this is about tones and pronunciation, for example English speaker always pronounce "hao" like "how", actually it's different but we can understand it.

There is a reason why people from south China can't speak Mandarin well. I think it's all about dialect, they have a variety of dialects while people from north China just speak a dialect similar to Madarin.

So Mandarin is a tool for us to communicate together because some dialect is hard to understand.

Today we speak dialects with family and local people. 

It's a white lie. Sometimes you get that after you said 你好. But fact is that many Chinese have problems with Beijing style Mandarin and pronunce it rather poorly. For quite a large number Mandarin is their second language. But as long as one can communicate it doesn't matter.