@Jason Starr: Thanks! Just to clarify: Since $X \times_{Y}U$ is an algebraic space (by the definition of an algebraic stack), we can consider the affine scheme from where there exists an etale surjection on the algebraic space and then show that this affine scheme is Noetherian. Now using your first comment the argument can be completed. I think this is what you meant. I wrote this just to be sure, since I am a beginner in algebraic stacks.

I guess I must mention this simple example was pointed out to me in stackexchange.

@Marc Hoyois: I don't see how is that true without any condition! For any non Noetherian ring $A$, consider the morphism $Spec(A) \longrightarrow Spec(k)$ where $k$ is a field. $Spec(k)$ is certainly Noetherian but $Spec(A)$ is not.

@Jason Thanks for your reply. I understand that the rings $\mathcal O_U(U)$ are reduced. However, I am trying to define torsion free sheaves in general by setting torsion submodule to zero. If $M$ is an $A$-module, then to define the torsion submodule of $M$, we need the ring $A$ to be an integral domain, not just reduced. It is in this sense that I wanted to ask how we can define torsion for sheaves on an integral stack.

@Jason: I tried but couldn't really understand why the sections of the structure sheaf being a reduced ring is enough to define torsion free sheaf on $ X_{et}$ (or stack) ? Since we don't have irreducibility, are you asking me to consider a cover by the irreducible components in some sense?Just to avoid confusion, let me mention the definition I am using for torsion free sheaf for the usual Zariski case. For $X$ integral, a sheaf $F$ on $X$ is torsion free, if $F(U)$ is a torsion free $O_U(U)$ module for all $U$.

@Jason...thanks! I got it. But now how to define a torsion free sheaf on the small etale site of a scheme ? As you have shown $\mathcal O_U(U)$ need not be a domain in case of $X_{etale}$, we can't use the same definition of torsion-free as in case of Zariski site.

@Jason...Sorry but I didn't understand what you meant by that case of disjoint union of copies of $X$. Could you kindly explain a little ?

Now if $\mathcal {O_X}(U \xrightarrow{etale} \mathcal X)$ is not an integral domain in general, then by the usual definition of a torsion-free sheaf, the structure sheaf wouldn't be torsion-free. But it seems fair enough to ask if we have the structure sheaf torsion-free while considering an integral stack...because this happens in case of an integral scheme.

I am asking about those 1-morphisms that are represented by etale maps which are not open immersions. The ring of global sections $\Gamma(\mathcal X_{etale}, \mathcal {O_X})$ of the small etale site of $\mathcal X$ is shown to be an integral domain in the Stack Project link I have shared. But I don't know if $\mathcal O_X(U \xrightarrow{etale} \mathcal X):= \mathcal O_U(U)$ would also be an integral domain since being integral is not an etale local property of schemes.

Thanks! I was actually studying quasi-coherator from Thomason-Trobaugh. However, it doesn't give any idea about coherator on sites. I just thought perhaps some of you know a source where the generalization is done. Now I am trying to generalize it myself. But you see unlike schemes there is no concept of affine cover on a site ! And the way Thomason-Trobaugh has constructed the coherator, uses affine cover. So may be I have to find some sort of replacement.