Tesla would be entitled, as any electrician is, to say that the energy is carried by the current. But he fails to appreciate that the current will only flow in conjunction with the associated field. The current will not flow without the field, because by definition the field is that thing which applies a force to a particle through the particle's property of charge. Short of proposing a fundamental new interaction involving charge (which he doesn't seem to be doing) Tesla's proposals can only be based on a fairly elementary failure to appreciate just what charges, currents and fields are all about.
This is, indeed, a rather common misconception, and often occurs amongst those trained in electronics and electrical engineering. It is natural (and perfectly reasonable within those domains) to think of the energy as flowing through wires, carried by currents. Voltage is seen as a kind of pressure and current is regarded as a flow of 'electricity', much like a flow of water in a pipe. Fields are regarded as incidental, sometimes a nuisance, and electricians are happy enough with any circuit until the wires reach an antenna, at which point a mysterious thing called radiation happens. The electrician's explanations tend to disolve into handwaving at that point as he struggles to address the question of just where and how is the energy transferred from the circuit to the field.
The physicist sees things differently by viewing nature from a wider perspective - one which isn't constrained by the limited and artificial view of nature given by circuit theory. When looking at an electrical circuit, the physicist sees a complex pattern of E and H fields formed into a rich and marvelously functional 3-D pattern by the boundary conditions imposed by lots of carefully arranged conductors and dielectrics. The technology of arranging those conductors and dielectrics is called electronics. Here and there those conductors might be formed into shapes that allow the fields to spread out in certain useful ways, and the electrician would call one of those bits the antenna.
The physicist's view of an electronic circuit as an intricate pattern of ripples and tensions in the EM field, anchored in place by a beautifully crafted assemblage of materials, is a wonder to behold and helps us to remember that no change in the physics occurs when we move from 'energy in the circuit' to 'energy in the air', and that such a move is merely a convenient change of descriptive language.
By the way, note that it is not necessary for us to say that the current 'causes' the field, or that the field 'causes' the current. There's no cause and effect process in EM interactions at the fundamental level. EM doesn't contain an 'arrow of time'. It is only when thermodynamics steps in, and you start to look at the so called entropy of the system that it becomes meaningful to say that energy flows from A to B. As far as the particle/field is concerned it is just A interacting with B. It is only when you move outside EM and consider the thermodynamic implications of the A-B interaction that you can begin to identify a cause-effect relationship. Only then can you unambiguously declare one to be the 'source' and the other to be the 'load'. The important thing is that charge and field are inseparable, you can't choose between one or the other as a choice of energy transfer mechanism, but you can choose between them as descriptions of the transfer process, and of course, both give the same correct answers.
