Note: although I will write 'text', the same applies to any image, code, or other resource that needs to be replaced.
Generalizing an answer is somewhat difficult since so much is game-specific. In general though you'll see virtually every translation project will try to fit as much text as is needed in, necessitating compressing data and/or extending the ROM. Dwindling the whole script to fit within space would only really be concidered if extension or further compression isn't possible. That said, shortening up lines for printability (ie. what will fit within a given window) is somewhat common.
The biggest factor is how the game was coded and your hardware target, though the ability of the coders inserting the script can be a factor. If they are incapable of the ASM work needed to redirect resources, or a full codec can't be written for a certain format, that will certainly affect which route they take.
There are really only two ways to replace resources. Either you tack new data at the end of the existing ROM and direct pointers/code to access it, or you physically replace the original data and shift other data as needed.
Obviously the first will extend the ROM by leaps and bounds, so it is fine for the more common situation of relatively general hardware concerns and no real limit on space.
The second will usually be more involved to program if you need to shift other resources around--at the extreme effectively rebuilding the ROM. In the same breath, you can say it can be remarkably easier since no redirection is involved if you stay within the original filesizes. However, if your game happens to use a very specific hardware arrangement and can't be extended, or to avoid adding an extra layer of compression, or just to avoid having to redirect pointers you may have to go this route. In general, you'll see more text dwindling with this arrangement to fit to size concerns. Again, matters on the case since two-byte japanese code means double the string lengths in ASCII.
In the end though, it usually isn't hardware concerns that will drive you to do something a certain way, but more what you're shooting for within the translation.
As I mention though, this really is a game-by-game basis. For instance, if the game happens to use a computing codepage for their japanese where every character is 2+ bytes long, the english equivalent might fit fine--especially if they stay away from kanji. In other words, due to a technical aspect of the game you can avoid a technical solution. However, that would only really apply to a subset of games from gen 5 consoles onwards.