The key concept here is the 1996 spelling reform — and contrary to many other issues it introduced, the ss/ß question has remained constant throughout all later modifications of the rules. In the 19th century, there were two different standards on when to use ß and when not. One tradition, founded by Johann Christian August Heyse, bases the choice on phonologic arguments, while the other, traced back to Johann Christoph Adelung, chooses based on typographic and morphological arguments.
None of the following discussion applies to Swiss (standard and spoken) German. The German-speaking community of Switzerland stopped using ß already in the early 20th century. In Switzerland, ß is never to be used except maybe for proper nouns originating in Germany/Austria/Luxemburg/South Tyrol. Instead, ss is always used.
Before the 1996 reform, the Adelung spelling (German article) was used. It dictated that ss should be used only between vowels that belonged to the same stem and if the preceeding vowel is short, while ß is to be used before consonants and at the end of a morpheme. This meant that similar words with very similar pronunciation were written differently based on the question of whether the /s/ sound was at the end of the word or not. For essen, it is mostly the difference between wir essen and ihr eßt that jumps to mind — the first two phonemes are pronounced identically yet the spelling differs. According to Adelung, essen is spelt:
Adelung’s spelling makes a lot more sense in the context of blackletter typesetting, whence the ligature ß originally developped from.
Since 1996, the Heyse spelling (German article) is in use. As mentioned, it bases the spelling on a phonologic argument alone, namely that a short vowel should be followed by a double consonant as is true for most other consonant sounds (the di- and trigraphs ch and sch being the most prominent exceptions). With the Heyse spelling, the pronunciation of the vowel (short or long) can thus immediately be deduced. And according to that spelling, essen is spelt:
Note how all the vowels are short and the ss spelling is constant.
Also Note that the past tense of essen is unaffected. It is always aß and derived — the a sound is long.
In blackletter type, the choice of spelling was more or less arbitrary; the question boiled down to whether there should be a ligature or a combination of ſ and s in a specific word. However, in antiqua typefaces a potentially important distinction may be lost when switching from Adelung to Heyse (while others are gained): the word Fassende can be interpreted as Fass-Ende in Heyse spelling while that would be Faßende in Adelung spelling. (The example is highly artificial though; the two words differ in more.)
Another verb whose spelling changed with the reform and which does not feature stem vowel variation is küssen so it may be a better example:
Ich küsse (both)
Du küßt (Adelung)/küsst (Heyse)
Er küßt (Adelung)/küsst (Heyse)
Wir küssen (both)
Ihr küßt (Adelung)/küsst (Heyse)
Sie küssen (both)
The ü vowel is pronounced identically in all cases.
Native speakers from Germany and Austria will either consistently use the Adelung spelling if they are more than approximately 30 years old, did not learn the new spelling at school and never bothered to adopt it later, or they will consistently use the Heyse spelling if they are younger and learnt it at school or actively decided to transition between them after the reform. If someone uses ißt in one sentence but isst in the next, everybody in Germany and Austria will consider that an error.
As mentioned above, Swiss will consistently use ss all the way through, but will possibly recognise correct or incorrect usage of ß if they use German or Austrian written media a lot.