Magnesium blocks are connected to iron structures to serve as cathodic protectors. A key element is that the magnesium should react slowly enough (i.e., selectively) to last for a long time (years), while providing enough electron flow to keep the iron negative in the midst of all its electron acceptors (oxidizers). If the iron were in a severely corrosive situation and you needed more current, and you think of using sodium as a sacrificial anode to provide a faster reaction, it wouldn't be practical for many reasons.
The sodium situation is where the electrons wouldn't bother traveling to the iron cathode; things are happening too fast because of the energetics of sodium reacting with water (forget the acid!). This situation does not permit the selectivity to form separate anodes and cathodes.
Dry cells are constructed of zinc anodes and carbon cathodes; impurities in the zinc surface can short-circuit the cell: the zinc becomes both anode and cathode, and is consumed inefficiently. This condition was ameliorated by amalgamating the zinc surface with a tiny bit of mercury. Mercury has a high hydrogen overpotential (i.e., it takes a higher potential to force enough protons to hydrogen atoms and then to molecules and then to bubbles which rise up.) The mercury prevented the initiation of local cathodes, and did not prevent the migration of zinc to its surface and thru it. So the selectivity was improved by the mercury amalgamation, allowing the zinc to be a purer anode. (The next step was to improve it further so as to eliminate the mercury!)
A single iron nail can have electrochemical corrosion all over its surface with anodes and cathodes that switch around and/or are so small that you can't distinguish them. You see only a rusty nail. One way to separate the anodes and cathodes and to make them visible is to put the nail in an agar gel with a little NaCl, phenolphthalein and potassium ferrocyanide. The nail begins to corrode at a random spot. This first spot will lose Fe++ ions to the gel, where they will react with ferrocyanide ion to turn blue. Electrons from the reaction will flood the iron nail; somewhere else, O2 will be reduced to hydroxide ions, which turn the phenolphthalein red. Typically, the head and point of the nail will become anodes (because they are more highly stressed, therefore a little more anodic) while the shaft will be cathodic. Pieces of rebar were more random in their reactivities.
So, oxidation and reduction can occur at the same electrode; it's just more illuminating (and useful) to see them separated instead of all mixed up.