It's more likely to stop in my wall than go through.
Based on tests I've seen, a typical JHP will go through multiple residential walls just fine, usually without opening up at all. Expansion and controlled penetration in the intended sense typically doesn't happen with harder materials unless the material is so hard that a FMJ or plain lead bullet would also flatten out (seen that, too). JHPs only work properly when they encounter a soft, wet material such as flesh--only then are they less likely to overpenetrate. This is actually a good thing for those who may need bullets to penetrate barriers before hitting concealed chunks of flesh that are up to no good.
In general, the type of bullet one chooses depends on the caliber, the target, and the intended effect (e.g. depth of penetration, width of the cavity created, etc.). For self-defense against humans and other critters, adequate penetration is probably the most important factor, followed by the width of the permanent cavity, and finally secondary effects such as the temporary stretch cavity. If a caliber is on the small side for a particular critter, then FMJ bullets might be necessary to achieve sufficient penetration, and would therefore be the optimum choice. If a SWC can penetrate enough, then it might be preferable as it tends to crush the flesh in its path more (a wide meplat seems to help).
The purpose of JHPs is to transform excess penetration into extra permanent cavity width, which is a good thing but only when there actually is excess penetration. If forced to defend myself from a bear or moose with a "puny" .40 S&W or .45 ACP handgun, for example, I'd use a solid, heavy, and physically hard type of bullet. At least then I'd have a half-decent chance of getting penetration to vital tissues--with JHP, I might as well turn the gun on myself (kidding
). For defense against humans (loosely speaking), I prefer at least 14" of penetration because some people are quite big these days, and I'll take whatever expansion I can get along with that, which is always a trade-off; given the option, I'd rather slightly overpenetrate than underpenetrate.
Nothing I've said here is absolute in my own mind, by the way--admittedly it's what I've gathered mostly from reading from a wide variety of sources, and hard, reliable data is difficult to come by for most people.