This requires drape. If you knit the shawl too stiffly, it won’t be able to wrap fluidly around your shoulders. The stitches need to be loose enough to allow the fabric to flow. To do this, experiment with a larger needle size, which will increase the drape. Using the shawl’s own weight also requires knitting a shawl that is big enough to drape effectively.
Shawl pins are great for heavier shawls or thicker shawls. However, shawl pins can sometimes be too substantial for really delicate shawls. When, in doubt, test your shawl pin at the corner of your shawl to see whether the stick part goes through the fabric easily and whether the whole apparatus is too heavy for your shawl.
This is a particularly effective strategy for shawls that are really lightweight or knit at a very fine gauge. With vintage brooches, you also have a finer pin. These are meant to go through finely woven fabrics, not just knits, so they will be much sharper and narrower than a shawl pin.
This is an especially common strategy for rectangular wraps, but it also works for triangular shawls or other shapes. Simply take the longest part of your shawl, wrap it around your shoulder, and tie the pieces together in front. There are different ways to tie the shawl, just as there are different ways to tie a necktie, and you can find lots of fun tutorials online.