Assuming we're dealing with 2d particles, you can just assign particle texture as you would normally, then create new shader material (under CanvasItem section of the Particle2D node) and paste this shader code:
shader_type canvas_item;
render_mode blend_mix;
uniform float shiny_scale = .5;
void fragment(){
const vec2 half = vec2(.5, .5);
COLOR = texture(TEXTURE, UV) * COLOR;
vec2 uv_scaled = clamp( (UV - half) / shiny_scale + half, vec2(0.0, 0.0), vec2(1.0, 1.0));
vec4 t = texture(TEXTURE, uv_scaled);
COLOR = mix(COLOR, t, t.a);
}
This will draw the texture twice, once normally modulated and once white and scaled over the modulated texture. Note that shiny_scale property will appear under Shader Params when the shader becomes active. Changing it will scale the white particle.
In order for this to work properly all border pixels of your texture must be fully transparent.