\(n\)-Bell groups satisfy the law \([x^n,y]=[x,y^n]\) while \(n\)-Kappe groups satisfy \([[x^n,y],y]=1\). The set of right two-Engel elements of \(G\) is denoted by \(R(G)\). Brandl and Kappe have shown \(G^{n(n-1)}\subseteq R(G)\) for \(n\)-Bell groups \(G\). The authors give a bound for the exponent of \(G/Z_2(G)\) for \(n\)-Bell groups \(G\); it divides \(12n^5(n-1)^5\). Locally graded Bell groups (i.e. \(n\)-Bell groups for some \(n\)) have a very restricted structure: The torsion elements form a locally finite subgroup and the quotient group is nilpotent of class \(2\). For Bell groups \(G\) in general it is shown that either \(G/Z_2(G)\) is locally finite or \(G\) has a finitely generated subgroup \(H\) such that \(H/Z(H)\) is an infinite group of finite exponent.