It is not so difficult, if you know how to express "each" in Japanese.
The common way to express "each" is using 「それぞれ（の）」 「[各]{かく}」, or 「ごとに」.
By using these words and ordering words correctly, you can remove ambiguity, as follows:
それぞれの人に本を2冊ずつ下さい。 / それぞれの人に本を2冊下さい。
Please give 2 books to each person.
各ペアは3ポイントずつ受け取る。 / 各ペアは3ポイントを受け取る。
Each pair of people will receive 3 points.
それぞれの人がりんごとみかんを2つずつもらう。 / それぞれの人がりんごとみかんをそれぞれ2つもらう。
Each person will receive 2 each of apples and oranges.
Notice that in the first two examples 「ずつ」 is not crucial anymore because 「それぞれ」 or 「各」 is enough to express "each", though 「ずつ」 still helps us properly understanding the meaning of each sentences.
In the last example, we have two different "each," so we still have two words for each "each."
In the left Japanese sentence 「ずつ」 refers to "each of apples and oranges" and 「それぞれ」 is for "each person."
By the way, in the second example, I used 「ペア」 to express "pair of people."
In this case, if we avoid this and stick to 「2人」, it becomes less natural but the following may be possible:
2人ごとに3ポイントが与えられる。
Actually they have a bit different nuances.
Using 「ペア」 (or 「組」 etc.) sounds like they are a fixed team, but if we use 「ごとに」 here, it sounds like each pair was chosen just for the purpose of giving 3 points.
