The actions of the general nonabelian tensor products can be any actions; you just need $G$ to act on the underlying set of $H$, $H$ to act on the underlying set of $G$, and for the actions to satisfy the compatibility conditions.
In practice, though, the most common action is given by some kind of conjugation, especially when $G$ and $H$ are either equal or subgroups of some larger group; or by automorphisms, but they need not be. For example, Zac Thomas has considered the problem of computing all possible tensor products of cyclic groups, where the mutual actions are arbitrary (though compatible).
There is a close connection between nonabelian tensor products (particularly nonabelian tensor squares when the action is conjugation) and commutators. See in particular L-C Kappe's Nonabelian tensor products of groups: the commutator connection, Groups St Andrews in Bath, 1997, II, 447-454, London Math. Soc. Lecture Notes Ser. 261, Cambridge University Press, 1999.
In the case where $G=H$ and the mutual action is conjugation, you obtain the nonabelian tensor square of $G$. In this case, the connection to commutators is even stronger. There is a nice commutative diagram that connects the nonabelian tensor square, the nonabelian exterior square, and the second homology group of $G$: \begin{array}{cccccccc} & & & & 0 & & 0 \\ & & & &\downarrow & & \downarrow\\ H_3(G)& \longrightarrow& \Gamma(G/G') & \longrightarrow & J_2(G) & \longrightarrow & H_2(G) & \to& 0\\ & & \downarrow & & \downarrow & & \downarrow\\ 0 & \longrightarrow & \nabla(G) & \longrightarrow & G\otimes G & \longrightarrow & G\wedge G & \to & 1\\ & &\downarrow & &\downarrow & &\downarrow\\ & & 1 & & [G,G] & = & [G,G]\\ & & & & \downarrow & &\downarrow\\ & & & & 1 & & 1 \end{array} The maps from $G\otimes G$ and $G\wedge G$ to $[G,G]$ are the "obvious" ones: you map $g\otimes h$ to $[g,h]$. One can interpret $H_2(G)$ as the group that contains all relations satisfied by the commutators in $G$ that are not universally satisfied (see, e.g., Clair Miller, The second homology group of a group; relations among commutators, Proc. Amer. Math. Soc. 3 (1952) 588, 595, MR 0049191 (14,133c)). All rows and columns are exact.
The connection in the general case can be also be realized thanks to a construction of Rocco (and independently by Graham Ellis and Frank Leonard) which gives the nonabelian tensor product by first constructing a group generated by copies of $G$ and $H$, and realizing the nonabelian tensor product precisely as the subgroup $[G,H]$ sitting inside it. (Although the papers discuss the case of the nonabelian tensor square, they generalize easily; N.R. Rocco, On a construction related to the nonabelian tensor square of a group, Bol. Soc. Brasil. Math. (N.S.) 22 (1991) no. 1, 63-79, MR 1159385 (93b: 20060); Graham Ellis and Frank Leonard, Computing the Schur multipliers and tensor products of finite groups, Proc. Roy. Irish Acad. Sect. A 95 (1995) no. 2, 137-147, MR 16603783 (99h: 20084)).