The two expressions are not exactly interchangeable. They may be sometimes, but even in those case, the emphasis of the sentence may change. To begin with "it's a pity" expresses negativity. If you tell someone "It's a pity you're not here", they might feel guilty, or uneasy. If you say: "I wish you were here", they just feel desired and loved. This is just a brief explanation. So much could be said for any instance in which one would use one or the other, but it's better to analyze them individually.

When English speakers say, "It is a pity it is raining again" or "It is a pity you couldn't visit me", they are usually expressing genuine sadness. It is an expression that is usually said when there is a reason for the sadness. For example, "It is a pity it is raining today because I planned a family picnic!'. OR "It is a pity you couldn't visit me today because I planned a lovely meal!" I hope this helps

I heard they say also 'that's a shame you are not here' or some other phrases with 'shame', when they regret of something. Could anyone clarify in what cases 'shame' would be better to use?

Greetings Andrej, Yes! You can say: It's a pity you're not here. Let me know if you have any doubts on how to use the word "pity". Ricky,