In Ring Theory, does a 'power' of a morphism represent composition? Say there is a ring homomorphism, denoted by $\theta$. If the notes use the expression $\theta^2$, then are they referring to the composition of the $\theta$ homomorphism with itself?

There is no special notation for powers which indicates specifically that composition is the operation. You will have to depend on context.
For example, the set of continuous functions from $\Bbb R\to \Bbb R$ can be made into a ringmonoid with composition or pointwise multiplication, and in both cases you would use ordinary power notation for its elements.
If the context is unclear, you'll have to look for clues. As was already mentioned, if the domain and codomain are different, then that will eliminate composition in many of cases. If the codomain is not a ring, then pointwise multiplication does not make any sense.
If you are specifically looking at ring endomorphisms of a fixed ring $R$, then it is probably composition. While any two functions from $R$ to $R$ can be added and multiplied pointwise, the result is not necessarily a ring homomorphism if the inputs are.
