The energy balance is indeed an interesting problem. For a monochromatic plane wave the source is an infinite sheet of sinusoidal current.
It is not trivial but is straightforward to calculate the Poynting vector for this arrangement. When you do so you find that energy propagates away from the current sheet with equal power density on both sides of the sheet. When you further calculate $\vec E \cdot \vec J$ at the current sheet itself you find that the work done by the current is equal to the radiated power. So the conservation of energy holds.
Now, Maxwell’s equations are linear and translation invariant, so you can simply shift the current sheet some distance to get two current sheets. The total field from the sum of the two current sheets is simply the sum of the fields from each sheet.
However, although the fields add linearly, the energy is not linear. So you could take a current sheet which by itself produces waves with some given power density $P_1$ and a second sheet which by itself produces a power density $P_2$ and when you add them together you get waves with a power density $P\ne P_1+P_2$.
The key is to recognize that the two sources affect each other. If you calculate the work done by the first sheet you will find that $\vec E \cdot \vec J \ne P_1$. In other words, the presence of the second source changed the work needed by the first source to produce the same current.
Such sources are called coupled, and this coupling can be damaging to RF power amplifiers driving coupled antennas. The power density of the two waves is different from the sum of the original waves, but it matches the power produced by the coupled sources.