This answer relies pretty heavily on http://www.scholarpedia.org/article/Local_operator. Some material can also be found in Peskin and Schröder and Weinberg, but the above article is quite self contained and gives a good overview.

I think there is a misconception regarding different types of divergences in QFT's. Consider for example a renormalizable QFT of a scalar field \(\phi_0\), where this denotes the bare field. The standard procedure then is that you rescale/shift the fields, masses and coupling constants so that your Lagrangian is expressed in terms of renormalized (physical) fields, masses, coupling constants and counterterms. The counterterms are fixed by appropriate renormalization conditions. Then all Green's-functions of the renormalized fields \(\phi_R= Z^{-\frac{1}{2}} \phi_0\) at different spacetime positions are well defined. However that does not mean that for example \(\langle \phi_R(x) \phi_R(x) \rangle_0\) is finite (0 denotes the interacting vacuum here). As an example consider a non interacting theory of a scalar field (where no field renormalization occurs at all). \( \phi(x) \phi(x) \) is ill defined since for example the vacuum expectation value (VEV) diverges. A regular operator mimicking the \(\phi^2\) operator in the free case is the normal ordered version \(: \phi^2 :\) which can be written (schematically) as \(: \phi^2 : = \phi^2 -\mathbb{1} \Delta(0) \), where the propagator and the identity operator appear.

This procedure of normal ordering inspires the proper definition of local regular operators in any interacting QFT. For any operator \(O(x)\) that is a product of elementary (and already renormalized) field operators and their derivatives at the same spacetime point we can define an operator \(N_{\delta}[O(x)]\), which can be expressed as a finite sum \(\Sigma_l Z_l O_l(x)\) where \(Z_l\) are (usually) infinite constants, and \(O_l(x)\) are other local field operators built from elementary fields and their derivatives. In some special cases \(N_{\delta}[O(x)]= Z O(x)\), but in the general case other operators appear on the righthandside. This is then called operator mixing. \(\delta\) is a parameter related to the mass dimension of the operator.

The prescription on how to construct these renormalized versions of local operators requires additional conditions, called the BPHZ normalization conditions. All of the above is a highly abbreviated and simplified version of http://www.scholarpedia.org/article/Local_operator. This article also mentions why operators like the electromagnetic current \(J(x)\) which consists of 2 fermion fields does not require this procedure. Due to it being conserved/ gauge invariant we have \(N_{\delta}[J(x)]=J(x)\). I hope this helps a bit