By default, the Mapillary mobile apps capture images with the largest resolution, at 4:3 ratio. Depending on your device, one image takes roughly 2–8 MB, giving you about 1000–4000 images if you have 8 GB of free storage on your phone. The average file size on iOS devices for each image is about 2 MB; on Android devices, it is generally a bit larger so you may need an external SD card (you can change the storage location in the app settings).
If you have concerns about the space required, you can use the app settings to reduce the frequency of capture, or if you're on Android, then turn on the option to take images in lower resolution.
On Android devices, by default capture will be stopped when you have less than 10% free storage space left, or when your battery level is under 10%. The iOS app has no battery warning but will stop capturing at 1% free disk space by default. You can change these values in the app settings too.
Keeping a copy of your images
Images are stored in the app folder and deleted after upload. Images are never stored in your camera roll/album. The only way to view them is through the upload view in the app. This also means that if you want to keep copies of the images, you need to do this before you upload.
Note that the images and GPS info are saved in separate files during capture, and in the app are combined upon upload without you needing to do anything extra. However, if you plan to copy the images, you need to remember this.
For Android, try this article on Google Support to learn how to transfer files between your phone and computer. Remember to copy the GPX file (that saves the GPS data) over as well. You need to geotag the images yourself (see here for instructions).
For iPhone, there is a similar article on Apple Support about transferring files. Before you copy the images, go to the Upload screen and push the EXIF data button at the top left (see below)—that will add geotags to the images without uploading them.
Tip: in case the above links don't work for some reason, you can always type "transfer files from iPhone / Android phone to computer" into your search engine to find other tutorials.