For a connected locally $(G,s$)-arc-transitive graph $Γ$ with $s\geqslant 2$ and an edge ${v,w}$, determining the amalgam $(G_v,G_w,G_{vw})$ is a fundamental problem in the area of symmetrical graph theory, but it is very difficult. In this paper, we give a classification of $(G_v,G_w,G_{vw})$ in the case where the vertex stabilisers $G_v$ and $G_w$ are faithful on their neighbourhoods, which shows that except for the case $G_v\cong G_w$, there are exactly 16 such triples.