Type Ibn supernovae (SNe) are a mysterious class of transients whose spectra exhibit persistently narrow He i lines, and whose bolometric light curves are typically fast evolving and overluminous at peak relative to standard Type Ibc SNe. We explore the interaction scenario of such Type Ibn SNe by performing radiation-hydrodynamics and radiative-transfer calculations. We find that standard-energy helium-star explosions within dense wind-like circumstellar material (CSM) can reach on day timescales a peak luminosity of a few 10 44 erg s −1 , reminiscent of exceptional events like AT 2018cow. Similar interactions but with weaker winds can lead to Type Ibc SNe with double-peak light curves and peak luminosities in the range ∼ 10 42.2 to ∼ 10 43 erg s −1 . In contrast, the narrow spectral lines and modest peak luminosities of most Type Ibn SNe are suggestive of a low-energy explosion in an initially ∼ < 5 M helium star, most likely arising from interacting binaries, and colliding with a massive helium-rich, probably ejecta-like, CSM at ∼ 10 15 cm. Nonlocal thermodynamic equilibrium radiative-transfer simulations of a slow-moving dense shell born out and powered by the interaction compare favorably to Type Ibn SNe like 2006jc, 2011hw, or 2018bcc at late times and suggest a composition made of about 50% helium, a solar metallicity, and a total ejecta/CSM mass of 1-2 M . A lower fractional helium abundance leads to weak or absent He i lines and thus excludes more massive configurations for observed Type Ibn SNe. Further, the dominance of Fe ii emission below 5500 Å seen in Type Ibn SNe at late times is not predicted at low metallicity. Hence, despite their promising properties, Type Ibn SNe from pulsational-pair instability in very massive stars, which requires low metallicity, have probably not yet been observed.