Secondary (also known as Johari-Goldstein or β-) relaxations are an intrinsic feature of supercooled liquids and glasses. They are important in many respects but the underlying mechanisms are not well understood. A long-standing puzzle is why some glasses show β-relaxations as pronounced peaks, whereas others as unobvious excess wings. Here we demonstrate that these different behaviours are related to the fluctuations of chemical interactions by using prototypical systems of metallic glasses. A general rule is summarized: pronounced β-relaxations are associated with systems where all the atomic pairs have large similar negative values of enthalpy of mixing, whereas positive or significant fluctuations in enthalpy of mixing suppress β-relaxations. The emerging physical picture is that strong and comparable interactions among all the constituting atoms maintain string-like atomic configurations for the excitations of β-events and can be considered as the formation of molecule-like metallic glasses.