Translation:They are on good terms with their neighbors.
Technically, both the singular and the plural would work, but I would be far more likely to say, "I have a good relationship with my neighbours." I think you would rarely, if ever, hear, "I have good relationships with my parents." It would be, "I have a good relationship with my parents." Maybe it is considered a single relationship because the people in question are being considered as a single unit. I can see how it would seem to make no sense to people for whom English is not their first language. I had honestly never given it any thought until today. That's just the way it's said. Anyway, there's my two cents, for what it's worth. :)
Perhaps "I have good relationships with my neighbours" would make more sense if you're referring to multiple different neighbours, Say those left of your house, Those right of it, And those across the street from it, While if you were just referring to one household of neighbours, No matter how numerous, You could use the singular "I have a good relationship with..."?
I disagree with this statement. "They have a good report with their neighbors" and "They are in good rapport with their neighbors" are both totally valid, natural sentences, at least in American English, and should be accepted translations. Neither is accepted as of today.
While similar in meaning, That seems to have slightly different implications to me, "A Good Relationship", Or "Good Relations" would more just imply you get along well with them, You respect them and are nice to them, And they do the same to you, While "In good standing" I feel is more one sided, And would usually imply the one one is in good standing with is higher ranking than them, You could be in good standing with a king, Or your boss, But you wouldn't really be in good standing with your neighbours.
As far as I can understand it, Italians use their determiners in many places we don't in English (and vice versa) so "i loro" means "the their" but since we don't say "the their" in English, it means what we say by just "their". Now, if I am wrong I hope a native Italian speaker corrects me.
There are many ways you could translate this, "They are on good terms with their neighbours", "They have a good relationship with their neighbours", "They have good relations with their neighbours", And probably more, All of which being different ways of saying effectively the same thing in English. Additionally, As "Sono" is the conjugation for both First Person Singular and Third Person Plural, You can substitute 'They' and 'Are' in those translations with 'I' and 'Am', Respectively.