We discuss a transmission problem driven by the $p$-Laplace operator, equipped with a natural interface condition. Two aspects of the problem entail genuine difficulties in the analysis. First, it lacks representation formulas. Also, its ellipticity may collapse as the gradient vanishes. Our arguments circumvent those difficulties and lead to new regularity estimates. First, we prove local boundedness for the solutions. Then we establish an integral estimate for the gradient in $BMO$ spaces. The latter implies solutions have a borderline Hölder modulus of continuity.