What does a non-deterministic guess "look like"?
Nondeterministic guess is a non formal term we use for two (or more) possible transitions from the same configuration. When talking about a non deterministic finite automata, if a state $q$ has two different transitions for the character $a$, and I want to show that it accepts a certain word $w$, then I treat the junction at $q$ where $a$ is read as the location where the &quot;guess&quot; happens. By &quot;guessing correctly&quot; I refer to the transition that will lead to the eventual acceptance of $w$ (whereas I don't care what would have happened if I took the other transition, the &quot;wrong&quot; guess).
One implementation of guessing a partition of $w$ to $w_1w_2$ would be to have two transitions from the initial state for each character read by the machine head. One transition goes to the state $q$, where $q$ refers to $w_1$ as whatever lies to the left of the current position of the head and $w_2$ as the rest. The other transition stays at the initial state and keeps moving right until it meets blank. Although it's important to realize how to implement such a &quot;guess&quot;, it is arguably more important to develop the intuition around the power of nondeterminism and understanding the such a nondeterministic  partition can be done (without going through the tedious details and elaborating on the transition function).
For your problem non determinism doesn't really make things much easier, as you could simply execute both machines on every possible partition (of which there are $n$ where $n$ is the input's length) in parallel via dovetailing.