Why do a pair of interactive machines have a common input? I know that an interactive proof system $(P,V)$ is a pair of interactive machines for a language $L$ if $V$ is polynomial-time and the following two conditions hold:

1) Completeness: For every $x \in L$,
$$\Pr [\langle P,V \rangle (x) = 1] \geq \frac{2}{3}$$

2) Soundness: For every $x \not\in L$ and every interactive machine $B$,
$$\Pr [\langle B,V \rangle (x) = 1] \leq \frac{1}{3}$$

Actually, in an interactive proof system $(P,V)$ for language $L$, $P$ proves the fact whether $x \in L$ to $V$.

However, for a general pair of interactive machines, what is the meaning of the common input? Can we construct a system $\langle A,B \rangle$ such that $A$ wants to send $x$ to $B$ and prove that $x$ is not malicious before sending it. Or can we construct other communication systems instead of a proof system?
Interactive Turing Machines are used to model real-world parties that interact and perform local computations using some algorithms. So, why would they not have a common input? The common input simply capture the fact that both parties can start a protocol knowing some common value, which perfectly makes sense in any real-life application. In the definition of interactive proof system, we want to abstract out what it means to interactively prove a statement; so, the natural approach is to let both parties initially know the statement, and ask for one of the parties to prove it to its opponent. The question of how the statement became known to the parties is irrelevant to the question of building an interactive proof.

That being said, yes, you can trivially let one of the parties send the statement to the other party, and then prove its truth with an interactive proof system. Depending of the specific interactive proof system you consider, some might not require knowing the actual statement during the interaction (although it will be clearly needed by the verifier when executing his verification algorithm on the transcript). In this case, the proof could indeed be performed first, and the actual statement be sent later. But that's not the case of all existing proof systems.