Are standalone statements conventionally considered to imply truth? From what I understand, the statement $\exists x(p(x) \vee q(x))$ in the English language sounds something like this: "There exists $x$ such that $p(x)$ or $q(x)$". But this sounds like an incomplete claim; "There exists $x$ such that $p(x)$ or $q(x)$ what? Are true? Are false? 

Logical formulae like this are always meant to be true, and if what they express is false, so is the formula.
That is, the formula you gave essentially says "It is true that there is an x which is p or q", but since the "It is true that..."-part is already part of the semantics of logical formulae anyway, you wouldn't explicitly spell out this when translating predicate logic into natural language - just like in natural language you don't say either "It is true that the sun shines" but simply "The sun shines."  
The reason why "... such that p(x) or q(x)" sounds incomplete may also be due to "p(x)" and "q(x)" not actually being full sentences with a verb etc. as we would expect but just abbreviations; if we substituted "p" and "q" for e.g. to "dog" and "hungry" (and the choice of variable and constant names is irrelevant to the semantics of the formula, as long as we stick to a unique interpretation of them), we could translate the formula into "There exists an x such that x is a dog and x his hungry", which doesn't sound incomplete at all.
So the reason why the formula may sound like an incomplete term to you probably is due to the poor level of translation into natrual language sentences (wording it as "such that p(x)", which isn't a "complete" sentence in sense of lacking a verb etc., instead of wording it e.g. "such that x is p"), rather than the formula itself being incomplete.
The default assumption is always that what the formula expresses is true, and if this is not the case, so is the formula.
