"multifaceted responsibilities" is a correct and usable phrase in written English.
You can use it to refer to a situation in which there are several, complex responsibilities to be fulfilled. For example, you could say "This project comes with multifaceted responsibilities, including providing regular updates to the team and managing a budget."
In other words, poverty in America is a multifaceted problem, and it needs a multifaceted response.
The power of expression is multifaceted and comes with responsibility.
Recognizing that public safety is a multifaceted challenge and therefore the responsibility of not just the police but a whole range of agencies and services will require a significant shift in police culture and management.
Responsibility embodies the multifaceted roles that fathers/male partners play in financially supporting the child from birth [ 23, 25].
Apart within some Think-Tanks or specialised future-oriented organisations, very few individuals have the opportunity or the responsibility to follow the multifaceted and substantial changes in the social, ethical, technological and cultural environment, and the possible educational implications.
This is compounded by the fact that exposure data is multifaceted and complex and seldom, if ever, the responsibility of any one organization to collect and maintain.
Community involvement in health services provision thus promises multifaceted benefits; including encouraging the community to take responsibility for help-seeking as well as sensitising them to health workers day-to-day working constraints.
Effective collaboration also permits an identification of associations among multifaceted stakeholders providing basic societal functions based on recognized roles and responsibilities.
In HuffPost Black Voices latest episode of "The Tanning Effect," the multifaceted Grammy Award-winner elaborated on how her fans have influenced her feelings of social responsibility.
A multifaceted song from a multifaceted artist.
