Peer support plays a vital role in expanding access to mental health care by providing empathetic, community-based support outside formal clinical systems. As digital platforms increasingly mediate such support, the design and impact of these technologies remain under-examined, particularly in Asian contexts. This paper presents findings from an interview study with 20 peer supporters in Singapore, who operate across diverse online, offline, and hybrid environments. Through a thematic analysis, we unpack how participants start, conduct, and sustain peer support, highlighting their motivations, emotional labour, and the sociocultural dimensions shaping their practices. Building on this grounded understanding, we surface design directions for culturally responsive digital tools that scaffold rather than supplant relational care. Drawing insights from qualitative accounts, we offer a situated perspective on how AI might responsibly augment peer support. This research contributes to human-centred computing by articulating the lived realities of peer supporters and proposing design implications for trustworthy and context-sensitive AI in mental health.