While trust means a win for quality, productivity and team spirit, it is also a loss. A loss of control for the coordinating forces. Once you start giving responsibility to other people - by nature you are loosing control over the end result.
I believe that one of the main reasons that leaders have difficulties to give responsibility to people who have more time, are more proficient or have the right tools to define the end-result. When I started working as a team-lead, I wanted everything we do within our team to be great. I had high ambitions and wanted really good results. The mistake that I made back then, is that I never defined the definition of checkpoints and alignments. I continuously gave feedback to all members of the team, whenever they liked it, and the amount of detail of the feedback often touched base strongly with possible solutions.
Of course, providing solutions as feedback is easier. However, it is not empowering, and, more importantly, it will not lead to the full potential of teams. Instead, what is important when giving responsibility is to define a clear frame of the responsibilities you give. Formulating a clear statement is not always easy. However, when its there, results can be measured with it. By giving the employee an objective (and even better a way to measure the objective) - you allow them to learn and grow more vastly. Giving employees feedback on a higher level, allows them to put your feedback in a larger context.

To give responsibility also means to understand. To be aware of everybody's strengths and abilities. As you, as well as every other human, including your employees working on the tasks you forward them, strives for great quality, this factor is very important. Having transparency over your employees targets, questions, experiences and abilities creates a room that can be filled with fulfilling objectives. This way you can constantly push your employees slightly over their comfort zone, allowing them to grow - without making them feel lost or uncomfortable. This way you also can balance out tasks that might be below their capabilities.
Giving responsibility is the challenge of finding the sweat spot of challenging without overwhelming. Leading people means to understand their strengths, constraints, experiences, wishes, and abilities.
As leaders we should stop thinking about the details wherever we have people who have more abilities, greater ideas or time to work on the them. As leaders we are obliged to create greater targets and put them into the human context, that you are in. As leaders, we have the obligation to understand our employees and react to their individual characteristics and abilities. Sharing liability requires sharing a community. Giving responsibility requires taking the responsibility over every individual in your team.
Find a driver for everything