"I have bread, and you have salt."
Translation:Já mám chléb a ty máš sůl.
Because you are contrasting You and I, Ty a Já. So the ty needs to be there.
Why can you just say a ty sůl? Because the verb is the same so it can be elided in the second clause. In the other direction "I have (the) bread and you (the) salt." is accepted for the same reason.
Even the English sentence contrasts what you have and what I have. That is the stressing that we are talking about.
The Czech sentence you answered simply does not make sense to a Czech speaker. The pronoun need to be there to make sense.
I think you will have a very hard time to construct a natural Czech sentence with two different implied pronouns in different clauses.