My ideas on how the twins have received their powers all hinge on the fact that the stones communicate with each other in the comics. They are connected in some way when they are relatively close to each other, and they call out to each other as they are collected. Using that premise for the MCU allows the Mind Stone in Loki's scepter (I know this is still debated, but I think Thanos loaned this to Loki in order to come back with two stones) to have communicated with the Tesseract during Avengers, and the Aether when it was used in London while the scepter was underground in Europe.
I believe that Baron Strucker was continuing with the Red Skull's technology of harnessing this cosmic power, and that led to the deaths of many volunteers. Only after changing the parameters of the energy siphoning was he able to succeed in granting powers to the Maximoff twins. This tapped into the 'memory' of the mind stone, and imbued them with a small portion of the power contained in the Tesseract and Aether.
Quicksilver was gifted with a limited control over Space, explaining his super speed and the familiar blue/white blur.
Even more telling is how Wanda now has limited control over Reality itself, and how her Scarlet Hex powers resemble the red light given off by the Aether itself.
I think this opens up a huge doorway for the future of the MCU as it could introduce a character akin to Professor X if they managed to tap the power of the Mind Stone itself. It could also lead to Thanos needing to kill the twins in order to wield the full power of the Infinity Gauntlet, and would have established the technology necessary to explain how the Inhumans harnessed the power of their artifact to generate the Terrigen Mists. I'm not suggesting that the Inhumans will also be born from Infinity Stones, but if the ability to 'mine' these ultimate cosmic power cells, then the ability to harness other cosmic sources is immediately believable.
Let me know what you think in the comments, and continue to Make Mine Marvel!