That is something that most people never notice, but yes, your power meter will read higher power levels in that situation. That is actually normal.
On the CB side, many golden screwdrivers noticed that if they have a high SWR, that the power meter reads higher. This has been used by many to claim that they increased the output power of the radio that they didn't actually change anything. A scam more or less. This even happens with Bird watt meters, which are seen by many as the most accurate meters out there. Even though you have the appearance of power gained, appearance is all it really is.
On the ham radio side, if you have an antenna tuner that has a dual needle SWR meter, you will notice that as you get closer to an SWR match, not only does the reflected power go down, but the forward power does as well. This is normal.
So what causes all of this?
The answer is SWR itself. One way of describing what SWR is is a measure of how much power is movine one direction (forward) compared to how much power is headed the other direction (reflect).
So what happens when the reflected power gets back to the radio, or a tuned antenna tuner or some other properly tuned matching circuit? The answer is, it is re-reflected towards the antenna, and it is done in such a way that it adds to the forward power that is already there, and thus the power meter, reading more power, shows a higher reading. Yep, its really that simple.
Now, in the case of an SWR meter that is placed between an antenna tuner and an antenna. This area will always have a high SWR, and only in very rare cases will have an SWR match. So, the fact that SWR will always be higher at this point, and given what was stated above, a power meter at this location will read a higher power rating than a power meter put before a tuned antenna tuner.
Again, its all in the SWR and its effect on both forward and reflected power on a feed line.
Hope this helps.
Steve