We've been getting along okay without using any of the multitudinous conventions that exist for communicating after the opponents open 1NT, but it is starting to feel like a rather gaping hole in our system.
A convention called DONT -- Disturb Opponents' No Trump -- has the advantage of a charming name, as well as being elegantly easy to remember and deploy. In brief, when overcalling opponents' 1NT, a bid of a suit means you have two suits, with at least 5-4 length: the suit you bid plus another higher-ranked suit.
2C: clubs and another suit
2D: diamonds and a major suit
2H: hearts and spades
2S: just spades, not many points
If you have a single suit, double. Assuming the intervening opponent passes, your partner will bid 2C, after which you bid the suit you like, or pass if it's clubs.
Responses to a suit bid
Assuming the intervening opponent passes, the responder bids like so:
Pass: support for the suit bid
2 of the next suit up: wants you to name your second suit (or pass if this is the one)
2 of another suit: natural, showing 5- or 6-card length
Raise in your suit: likes it, with some strength, but non-forcing
2NT: strong hand, wants to go to game in your second suit. This is unusual.
Here's an example. After West's 1NT opening, North overcalls 2C, meaning he has length in clubs as well as a higher suit. South responds with the next suit up, meaning she doesn't favor clubs but has probably 3-card length in the other suits. North then bids his second suit, and the contract lands there.