mathematics

The expression—not equation—1 ± 1 in mathematics outside of statistics (where, in the context of confidence intervals, it might mean 0 to 2) is short for 1 + (±1), and equals 0 or 2 (it does not equal 0 and 2, because no number can simultaneously have two values). It is read aloud as “1 plus or minus 1”, but for the sake of brevity many of us further—both silently and aloud—shorten it as “1 plus minus 1” with no ill effect.

On a related note: the faux equation x = ±b is mathematical shorthand for the disjunction x=-b or x=b of two equations.

Since your first query's 1 ± 1 means 1 + (±1), all these ±1s are the exact same object, and are all read aloud as “plus (or) minus 1”.

Although it is rather common to verbalise ± 1 as “positive or negative 1”, this is not technically correct, and “plus (or) minus 1” is the logical and arguably correct verbalisation.

As prefixes, - and ± are unary (as opposed to binary) arithmetic operators: -(-1) (“minus minus 1”) just means to flip the sign of 1 twice. The outer - isn't indicating that -(-1) is negative, while the inner - is indicating neither that 1 is negative nor that 1 has both a positive and a negative version; thus, verbalising -(-1) as “negative negative 1” is neither instructive nor terribly coherent.

Elaboration here.