Do you use both my home and our home to your friends?
Solution 1:
I've heard Americans use, what is known as the 'Majestic Pronoun'; this "royal we/us" form in speech, but I never use it. Is this 'proper' usage? Sure. Such stuff is just a matter of personal preference in spoken and informal written English. I suppose it fits some folks' definition of being modest by not emphasizing themselves by saying "Send us a letter" instead of "Send me a letter", "Give us a call" instead of "Give me a call", etc.
The expression our mother, however, is another issue. This usage is only, really used by the king or queen. Most Americans wouldn't use this expression unless referring to the common mother of the speaker and listener or the common mother of a group (2 or more) siblings. Using the royal we/our here can sound pretentious for anyone but the king and queen.
The same goes for our home. I don't say home when I mean house or apartment (but that's another dispute), and I don't say our {house / apartment / place} unless I'm talking about where my family (wife & son) and I live. When I lived alone, I always said My {house / apartment / place}.
Solution 2:
Just a note to document Bill Franke's answer, which reflects my own experience of US usage:
OED notes this use of us for me as "dial. and coll." (ie, dialectal and colloquial). It offers an 1828 citation, but it's certainly older than that; the last verse of Robert Burns' Auld Lang Synte (1796) starts
And there’s a hand, my trusty fiere!
and gie's [ie, give us] a hand o’ thine!
But OED does not cite corresponding uses of we and our, only the "royal and imperial" and authorial or editorial uses.
Solution 3:
No, this particular colloquial use of us for me is restricted to the word us as an unstressed pronoun after the verb; we or our cannot be used in this sense.
In fact, all the examples I can think of have it as the indirect object of an imperative sentence: Give us a bit!, Show us your new car, Send us a note, or the syntactically similar benefactive: Cut us another slice!.
I don't think people use the construction with a direct object, as *Follow us or *Watch us.