Note: Please specify which OpenGL version you use when you post questions. That goes for any API.
When you want to render primitives with alpha blending, you can't have depth writes enabled. You need to draw your blended primitives sorted back-to-front. If you have opaque objects in the scene, render them first, and then draw your transparent primitives in a back-to-front sorted fashion with depth test enabled and depth writes disabled.
glEnable(GL_DEPTH_TEST);
glBlendFunc(GL_SRC_ALPHA, GL_ONE_MINUS_SRC_ALPHA);
glEnable(GL_POINT_SPRITE);
glEnable(GL_CULL_FACE);
while(1)
{
...
glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
glDisable(GL_BLEND);
glDepthMask(1);
RenderOpaque();
SortSprites();
glEnable(GL_BLEND);
glDepthMask(0);
DrawSprites();
...
}