Galaxy mergers provide a mechanism for galaxies to effectively funnel gas and materials toward their nuclei and fuel the central starbursts and accretion of supermassive black holes. In turn, the active nuclei drive galactic-scale outflows that subsequently impact the evolution of the host galaxies. The details of this transformative process as they pertain to the supermassive black holes remain ambiguous, partially due to the central obscuration commonly found in the dust-reddened merger hosts, and also because there are relatively few laboratories in the nearby universe where the process can be studied in depth. This review highlights the current state of the literature on the role of accreting supermassive black holes in local luminous infrared galaxies as seen from various windows within the electromagnetic spectrum. Specifically, we discuss the multiwavelength signatures of the active nucleus, its associated feeding and feedback processes, and the implications of multiple supermassive black holes found in nearby interacting galaxy systems for galaxy evolution from the observational perspective. We conclude with a future outlook on how the topic of active nuclei in low- and high-redshift galaxy mergers will benefit from the advent of next-generation observing facilities with unparalleled resolving power and sensitivity in the coming decade.