Duplicate solutions in NSolve

This is an extended comment. Actually, the answers only appear to be the same. For instance, eq[[6]] and eq[[7]] actually are

{{s1 -> 0, i1 -> 0, s2 -> 18.249999999999982`, i2 -> 0, s3 -> 0, i3 -> 0}}
{{s1 -> 0, i1 -> 0, s2 -> 18.249999999988777`, i2 -> 0, s3 -> 0, i3 -> 0}}

More generally, Union@eq does not reduce the number of solutions, indicating that there are small differences among them. Perhaps, NSolve performs its analysis on the equations in various orders, yielding solutions that differ in some cases only by roundoff.

Incidentally,

Solve[Rationalize[eqns, 0], unks]

gives the twelve desired solutions exactly.

$Version

(* "11.3.0 for Mac OS X x86 (64-bit) (March 7, 2018)" *)

Rationalize the equations

eqns = {0 == -0.07 s1 + 
       0.3 (0.75 i1 + s1) (1 - 0.05 (i1 + i2 + i3 + s1 + s2 + s3)), 
     0 == -0.28 i1, 
     0 == -0.07 s2 - 0.5 i1 s2 - 0.5 i2 s2 - 0.5 i3 s2 + 
       0.8 (0.75 i2 + s2) (1 - 0.05 (i1 + i2 + i3 + s1 + s2 + s3)), 
     0 == -0.28 i2 + 0.5 i1 s2 + 0.5 i2 s2 + 0.5 i3 s2, 
     0 == -0.07 s3 - 0.4 i1 s3 - 0.4 i2 s3 - 0.4 i3 s3 + 
       0.7 (0.75 i3 + s3) (1 - 0.05 (i1 + i2 + i3 + s1 + s2 + s3)), 
     0 == -0.28 i3 + 0.4 i1 s3 + 0.4 i2 s3 + 0.4 i3 s3} // Rationalize // 
   Simplify;

unks = {s1, i1, s2, i2, s3, i3};

Use arbitrary precision by setting the WorkingPrecision

eq = NSolve[eqns, unks, WorkingPrecision -> 7]

(* {{i1 -> 0, s1 -> 0, s2 -> -62.9200 - 11.5510 I, i2 -> -15.1800 + 38.4020 I, 
  s3 -> 79.3500 + 14.4388 I, i3 -> 15.2500 - 38.7566 I}, {i1 -> 0, s1 -> 0, 
  s2 -> -62.9200 + 11.5510 I, i2 -> -15.1800 - 38.4020 I, 
  s3 -> 79.3500 - 14.4388 I, i3 -> 15.2500 + 38.7566 I}, {i1 -> 0, s1 -> 0, 
  s2 -> 18.25000, i2 -> 0, s3 -> 0, i3 -> 0}, {i1 -> 0, s1 -> 0, s2 -> 0, 
  i2 -> 0, s3 -> 18.00000, i3 -> 0}, {i1 -> 0, s1 -> 14.21852, s2 -> 0, 
  i2 -> 0, s3 -> 0.700000, i3 -> 0.4148148}, {i1 -> 0, s1 -> 15.33333, 
  s2 -> 0, i2 -> 0, s3 -> 0, i3 -> 0}, {i1 -> 0, s1 -> 0, s2 -> 0, i2 -> 0, 
  s3 -> 0.700000, i3 -> 9.41499}, {i1 -> 0, s1 -> 14.30667, s2 -> 0.560000, 
  i2 -> 0.4666667, s3 -> 0, i3 -> 0}, {i1 -> 0, s1 -> 0, s2 -> 0.560000, 
  i2 -> 10.60545, s3 -> 0, i3 -> 0}, {i1 -> 0, s1 -> 0, s2 -> 0, i2 -> 0, 
  s3 -> 0.700000, i3 -> -1.714995}, {i1 -> 0, s1 -> 0, s2 -> 0.560000, 
  i2 -> -1.245448, s3 -> 0, i3 -> 0}, {i1 -> 0, s1 -> 0, s2 -> 0, i2 -> 0, 
  s3 -> 0, i3 -> 0}} *)

Verifying solutions

And @@ (eqns /. eq // Flatten)

(* True *)

NSolve returned the expected 12 solutions

Length[eq]

(* 12 *)