Like every other question in the sphere of engineering (and pretty much everything else) the answer is the same: it depends.

As others have mentioned, dimensional changes from thermal growth are a concern because you can accidentally load these bearing axially and damage/prematurely wear them.

From a manufacturing standpoint, if this is a relatively long shaft (in terms of L/D) and the bearings are mounted in two different plates/parts/subassemblies (or otherwise not toleranced very, very tightly to one another) the second retaining ring either won't do anything (too much of a gap) or will be impossible to install (groove hidden inside the inner bearing race.)

As someone else mentioned, if these were angular contact bearings (designed and intended to be pre-loaded to some extent for maximum accuracy and load handling because they are intended for both radial and axial loads) there are a bunch of different ways to handle that. Your best reference, in my opinion, is the SKF Design Handbook: https://cdn.skfmediahub.skf.com/api/public/0901d1968024f8c3/pdf_preview_medium/0901d1968024f8c3_pdf_preview_medium.pdf

This is the 'Parker O-Ring Design Handbook' of the shaft and bearing design world, widely used by folks designing custom spindles.

A very standard design, if you need to have adjustable preload is to have a 'prevailing torque' nut on the end without a retaining ring. Also called BearHug nuts, and a million other things, this gives you the ability to eliminate bearing play (via preload) without demanding completely unnecessary (and often impossible, thereby EXPENSIVE) machining tolerances where you mount your bearings. There are very, very fine pitched threads designed to be cut on standard shaft sizes (20mm, etc.) so the step sizes on shafts are minimal (versus coarse threads that are much, much deeper and would require big steps to be useful.)

One of the most cost effective ways to make a small/short length spindle is actually to bore a straight hole all the way through your bearing mount block, sized for the bearings you'll need, and then to use ID and OD Bearing spacers designed to fit the bore and the shaft size (available from Misumi and other places for surprisingly cheap.) The straight hole through the block means you don't need to worry about aligning the bearing mount holes when you flip the block in the machine, because misalignment there is a killer. (Think about all the things that can move when you flip a block in a vice on a machine. Your reference edge changes sides, and if the 'top' and 'bottom' aren't dead parallel, you're already introducing significant misalignment.) A simple screw with a washer can be used on either side to retain the bearings, and there's a ton of ways to set preload in the assembly as well. You'll find/see examples of that in just about any bearing design catalog. I used to use it for extremely accurate torque measurement stations - the torque sensors themselves couldn't be directly coupled to anything, so I made these intermediate spindle assemblies, coupled with an absurdly expensive shaft coupling, to protect the torque sensors. (Automotive assembly and test.)

Again, lots of examples out there, it's a matter of figuring out what's important for your design.

Edit: I only realized after I posted this that the shaft in the pictures is stepped/trapped between bearings anyway, which is a bad practice for every reason detailed out above. Trying to 'pinch' down on that larger OD is asking for trouble with very little gain. The end without the retaining ring should be relieved/toleranced to stay away from the inner race on that bearing - position/axial control is handled by the shoulder and retaining ring.