First of all, if you play Arena, you win/lose Elo like in any non-Arena game.
Then, you win/lose EAS points (EAS = Elite Arena Score) which work very similar to ELO. This EAS score is used to rank the players in Arena but your EAS score is hidden as long as you are not in Elite league.
Additionally, before you reach Elite, you are in different leagues (bronze, silver, gold,...) which have not much meaning at all. They simply are there to stimulate you to play on till you reach Elite. After each game you win/lose shields which help you progress to the next level. The number of shields you earn depends on the league levels your opponents are in. The shields you win/lose are completely independ on the strength (ELO, EAS) of your opponents - they only depend on the league levels your opponents are.
Since your question is about shields, the answer to your question lies in the 3rd paragraph. If your opponents are in much lower leagues than yourself you might win very few or none shields even if you win.
Since you still earn regular ELO and (hidden) regular EAS points even if you earn no shields, I have added the first two paragraphs to provide the context of the bigger picture.