Believe is a verb that can take three different types of Object Complement.
It can take a tensed Embedded Question complement:
- He believes [what you tell him].
It can take a tensed That-clause complement:
- He believes [that [the moon is made of white cheese]].
And it can take an Infinitive complement, with B-Raising:
- He believes [her to be the best shotputter in Indiana].
The only kind it doesn't take is the Gerund complement; this is ungrammatical:
- *He believes Shakespeare writing Canterbury Tales.
The two different sentences in the original question simply have different complement types. Both are grammatical, both mean the same thing, and either will get to the same place, though by a slightly different route. No worries, eh?
I believe it's valid has a That-complement. The complementizer that (which is optional here because it's an object complement) has been removed. Put it back (and uncontract the contraction while you're at it) and you get I believe that it is valid, which is totally unexceptionable, though a bit stuffy for online discussion. Which is why it gets shortened.
I believe it valid has an Infinitive complement. The original infinitive complementizer to and the auxiliary infinitive be have been deleted by the common rule of To Be-Deletion, leaving only the Raised subject it and the predicate adjective valid, the guts of the sentence, with all the auxiliaries stripped away. Put them back and you get I believe it to be valid, which is equally unexceptionable.
Deletions are common; check for empty slots. But you have to know what slots there are.