The answer can vary.
Let's assume you are tuned to a specific pitch reference. AFAIK A = 440 is the most common.
If you are going by sound, you could call it Eb or D#. When the tone is is the reference, it's called concert pitch. The name would change depending on the key your are in. Eb and D# are enharmonic, or the same pitch.
However, it's not all that uncommon for people to tune down and still think of it as an open E string. This can work fine as long as everyone in the band is detuned by the same amount. For example if the guitar and keyboard is also tuned down a half step, it's fine to call it open E. However if one of the other instruments is not detuned by the same interval, then it causes communication problems.
The resulting problem is actually common in ensembles because many instruments are what we call "transposing." For example Bb trumpet, Eb saxophone are transposing instruments. When a Bb trumpet plays a C it sounds like concert Bb. When an Eb sax plays a C it sounds like concert Eb. On your detuned bass, a C would sound like a B natural.
This is confusing, so hopefully I did it right
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To navigate this problem the players must know both the pitch native to the instrument as well as the concert pitch. Additionally, if the band has a conductor, then he/she must know the transposition intervals for all instruments. If I remember correctly the score is actually written in each instrument's native key.
In some cases a musician might choose to play an instrument in a different native key. For example a Tuba player might use a C instrument or a Bb instrument. AFAIK Tuba parts are normally written in concert key, so if the Tuba player uses a Bb instrument they will have to transpose (use different fingerings).
The modern French horn is even more complicated. I believe the modern horn essentially has two sides. One side is transposing to Bb and the other side F. It's sort of like switching between a Bb and F instrument. I believe the native key is considered F, but the fingering changes depending on which side of the horn is being used.
You may not know it, but bass guitar is actually a transposing instrument. When you play the open E string, which is written as E2, it sounds an octave lower E1.