Charcoal, 16 bytes

ＩＥ⌕Ａθ=⁻ι⁺κ⊗№…θι-

Try it online! Link is to verbose version of code. Explanation:

    θ               Input string
  ⌕Ａ                Find all indices of
     =              Literal string `=`
 Ｅ                  Map over indices
       ι            Current index
      ⁻             Subtract
         κ          Number of previous `=`
        ⁺           Plus
           №…θι-    Number of previous `-`
          ⊗         Doubled
Ｉ                   Cast to string
                    Implicitly print

The number any given = prints is equal to the number of +s so far minus the number of -s so far. The number of +s so far plus the number of -s so far plus the number of =s so far is of course the index of the = in question, so the number of +s plus -s can be calculated from the difference, and the difference between the number of +s and -s by further subtracting double the number of -s.

As @Jonah points out in his J answer, you can also translate the -=+ characters to the integers -1..1, and output the cumulative sums at the positions of the 0s. This also takes 16 bytes in Charcoal:

ＩＥ⌕Ａθ=ΣＥ…θ⊕ι⌕=+λ

Try it online! Link is to verbose version of code. Explanation:

    θ               Input string
  ⌕Ａ                Find all indices of
     =              Literal string `=`
 Ｅ                  Map over indices
         θ          Input string
        …           Truncated to length
           ι        Current index
          ⊕         Incremented
       Ｅ            Map over characters
               λ    Current character
            ⌕       Find index in
             =+     Literal string `=+`
      Σ             Take the sum
Ｉ                   Cast to string
                    Implicitly print

Charcoal's Find returns -1 when there is no match, which is what we want for -.

J, ^{29 26} 25 bytes

0(=echo"++/\@])'-='&i.-1:

Try it online!

-3 thanks to ovs!

-1 thanks to Neil!

Inspired by Neil's answer.

Consider +=+=:

• '-='&i.-1: Convert -=+ to _1 0 1

  1 0 1 0
  

• 0(=...+/\@]) Scan sum and filter only elements at the 0 positions:

  1 0 1 0  ...  1 1 2 2
  1 2                  NB. 1 and 2 are the values at the 0 positions
  

• echo"+ Print each value

  1
  2
  

Retina 0.8.2, 44 bytes

=
=¶$`=
-1=A`
%O`.
+`\+-

((-)+|\+*)=+
$2$.1

Try it online! Link includes test cases. Explanation:

=
=¶$`=

Prepend all of the prefixes of the input that end in an =.

-1=A`

Delete the original input. (Assuming it ended in an =, this will have been prepended as the last prefix.)

%O`.

Sort the characters in each prefix.

+`\+-

Cancel out the +s with the -s.

((-)+|\+*)=+
$2$.1

Convert to decimal.

C (gcc), 73 bytes

f(char*p){int a=0;p--;while(*++p)*p==43?a++:*p==45?a--:printf("%d\n",a);}

Ungolfed:

void f(char*p){
    int a=0; // Accumulator
    p--; // The loop increases the pointer so compensate
    while(*++p){
        if(*p==43)a++; // 43 and 45 are ASCII codes for + and -
        else if(*p==45)a--;
        else printf("%d\n",a);
    }
}

Try it online!

05AB1E, 10 bytes

΄+-„><‡.V

Try it online or verify all test cases.

Explanation:

Î           # Push 0 and the input
 „+-        # Push string "+-"
    „><     # Push string "><"
       ‡    # Transliterate all "+" to ">" and "-" to "<" in the input
        .V  # Evaluate and execute as 05AB1E code:
            #  `>`: Increase the top by 1
            #  `<`: Decrease the top by 1
            #  `=`: Print the top with trailing newline, without popping

Because = in 05AB1E does exactly what we want it to in +-=, it might be the perfect language for this challenge.