(1) Sara and Jack buy and use a car and a bicycle, respectively.
(2) Sara and Jack buys and uses a car and a bicycle, respectively.
(2) is ungrammatical: it can only be parsed as conjunct (and hence plural) subjects with conjunct singular verbs.
(1) is grammatical, but it doesn't mean what you intend. That respectively distributes the immediately preceding conjunct term, a car and a bicycle between the two subjects, so what it says is that
Sara buys and uses a car, and Jack buys and uses a bicycle
Since what you mean is that Sara buys a car and Jack uses a bicycle, you'll have to find another way of distributing the conjunct verbs, such as:
Sara buys, and Jack uses, a car and a bicycle, respectively.
Both of these are grammatically acceptable; but I urge you to restrict their actual use to homework assignments and examination essays where you want to demonstrate your mastery of old-fashioned syntactic ingenuities. The use of "respectively" and complicated distributions makes your sentences difficult to parse, which is a discourtesy to your reader; in many cases it doesn't even save keystrokes. Say things as simply as possible:
Sara buys a car and Jack uses a bicycle.