The short answer to this is that Luke isn't a slave. Although Owen has been brow-beating his nephew on an almost continual basis (to the point that he's withdrawn his application to the Imperial Academy), the simple fact is that he's old enough to make his own decisions and they certainly can't stop him if he wants to just get up and leave.
As we see in the film's official novelisation, Luke has a gentleman's agreement with Uncle Owen to remain for one more season. Evidently this isn't the first time that he's succumbed to their persuasion:
“You know,” Luke replied distantly, “I think these ’droids are going
to work out fine. In fact, I—” He hesitated, shooting his uncle a
surreptitious glare. “I was thinking about our agreement about me
staying on for another season.”
His uncle failed to react, so Luke
rushed on before his nerve failed. “If these new ’droids do work out,
I want to transmit my application
to enter the Academy for next year.” Owen scowled, trying to hide his
displeasure with food. “You mean, you want to transmit the application
next year—after the harvest."
“I need you here, Luke. You understand that, don’t you?” “It’s another
year,” his nephew objected sullenly. “Another year.” How many times
had he heard that before? How many times had they repeated this
identical charade with the same result?
As to why Beru would be more amenable to the idea, it boils down to three factors;
- His friends have already left for the Academy:
“Owen, you can’t keep him here forever. Most of his friends are gone,
the people he grew up with. The Academy means so much to him.”
- He's clearly unhappy at home:
“He never will be, no matter how hard you try to make him one.” She
shook her head slowly. “He’s got too much of his father in him.”
- There's no evidence that entering the Academy would be a threat
Note that they didn't even bother to change his name, nor is there any special evidence that the Empire is actively seeking him. Owen seems to think that all of that Jedi nonsense is behind them now.