Can we have mutual elementary embeddability between distinct transitive sets? Is it consistent with $\sf ZFC$ to have mutual elementary embeddability between distinct transitive sets?
Formally, is the following theory consistent? $${\sf ZFC} + \exists M \exists N: M,N \text { are transitive } \land M \neq N \land M \prec N \land N \prec M$$, and if consistent, then what's its consistency level?
Note: $M \prec N$ refers to existence of an elementary embedding (in the language of set theory) from $M$ to $N$.
Note: The same question has been posted as a revised question to an older one at MathStackExchange.
 A: There are examples of this where $M,N$ are also models of ZFC, in the following paper, which is joint with Monroe Eskew, Sy Friedman and Yair Hayut: https://arxiv.org/abs/2108.12355
Thus, you certainly get such a situation if $0^\sharp$ exists. (The examples constructed in the paper from $0^\sharp$ are with proper class models, but you can find some set sized restrictions of those examples.)
It's also easy to see that if there are $M,N$ proper class models of ZF/ZFC and elementary $j:M\to N$ and $k:N\to M$, then $0^\sharp$ exists.
However, your question asks about sets $M,N$, and only requires they be transitive (not be models of any particular theory).
The set-sized version (even if one demanded also some theory) does not imply $0^\sharp$ exists, because if it holds in $V$ then it holds in $L$, with $M,N$ countable in $L$, by $\Sigma^1_2$-absoluteness. If $M,N$ are transitive sets modelling ZFC, and there are elementary $j:M\to N$ and $k:N\to M$, then letting $\alpha=\mathrm{Ord}\cap M=\mathrm{Ord}\cap N$, we get an elementary $j\circ k:M\to M$, and this restricts to give an elementary $\ell:L_\alpha\to L_\alpha$. Conversely, if there is an ordinal $\alpha$ and an elementary $\ell:L_\alpha\to L_\alpha$, then the construction of the paper cited above produces transitive $M,N$ and elementary $j:M\to N$ and $k:N\to M$, with $M\neq N$. As I recall, Dmytro Taranovsky had an answer on MO regarding the consistency strength of an elementary $\ell:L_\alpha\to L_\alpha$, but I can't find that at present. However, for your actual question, $M,N$ are arbitrary transitive sets, and there I don't see immediately whether one even gets an elementary $\ell:\beta\to\beta$ for some ordinal $\beta$ (for example, $\beta=$ the rank of $M$?)
