'Non-competitive active site–binding inhibitors' are called mixed-type inhibitors. These inhibitors exhibit features of both competitive and non-competitive inhibitors, as they increase Km (like a competitive inhibitor) and decrease Vmax (like a non-competitive inhibitor).
What an interesting question!
In theory, a reversible inhibitor binding to the active site of an enzyme is per definition competitive. In a review by Blat (2010) the author mentions that active site–binding inhibitors that display non-competitive inhibition are indeed unusual. Such inhibitors are referred to as mixed-type inhibitors. In many cases the unusual behavior is observed with (1) enzymes utilizing an exosite for substrate binding, or (2) isomechanism enzymes, (3) enzymes with multiple substrates/products and⁄or (4) products and two-step binding inhibitors.
(1) Enzymes with an exosite have a substrate recognition site that is different from the active site. For example, some proteases bind their target on the exosite and then catalyze proteolysis in the active site. Mixed-type inhibitors then bind to the exosite, thereby inducing non-competitive behavior, as the active site is not bound.
(2) isomechanism enzymes are enzymes that undergo several structural transitional changes during catalysis. When one of these conformations is rate limiting and binds the mixed-type inhibitor, it may non-competitively inhibit the enzyme when another conformational form binds the substrate.
(3) Enzymes with multiple substrates or products sometimes follow a sequential binding and release of two substrates (or products). Suppose substrate A and cofactor B (e.g. NADPH). The enzyme converts substrate A into A' and cofactor B to B'. Now suppose that the enzyme obligatory has to release A' before B' can be released and that the B' bound state is unable to bind substrate A in the active site, but is able to bind the mixed-type inhibitor. In this case, again, a non-competitive inhibition pattern is observed.
(4) Two-step binding inhibitors refer to inhibitors that cause a conformational change of the enzyme after binding its inhibitor, that very slowly reverts back. In the conformationally changed state, the enzyme cannot bind the substrate and competition with substrate is lost. The most extreme case are inhibitors that covalently bind to the active site (note this is the mechanism addressed by @RoverEye).
You ask what the dose-response graphs are of mixed-type inhibitors. A more common way of displaying inhibitor behavior is using Lineweaver-Burk plots. Many enzymes follow Michaelis-Menten kinetics (Berg et al., 2002) and by plotting the enzyme kinetics as Lineweaver-Burk plots more insight is gained into the affinity (Km, you call this 'potency') and maximum rate of the enzyme (Vmax, you call this 'efficacy') compared to dose-response graphs. In Lineweaver-Burk plots the reciprocal velocity of the enzyme is plotted against the reciprocal of the substrate concentration, resulting in a straight line out of which affinity an maximum velocity can be easily obtained via linear regression.
The following graphs obtained from Illinois Institute of Technology show the three different modes of inhibition discussed, starting with the two common types:
Competitive inhibitors increase Km (i.e., decrease substrate affinity).
Non-competitive inhibitors decrease Vmax (i.e., decrease substrate turnover)
Mixed-type inhibitors increase Km and decrease Vmax
- Berg et al. (2002). Biochemistry, 5th edition
- Blat, Chem Biol Drug Des 2010;75(6):535-40