You do (at least as between you and Instagram—your employer may have ownership rights in certain situations)!  Instagram does not claim ownership of any content that you post.

You do grant Instagram very broad license rights:  a non-exclusive, fully-paid and royalty-free, transferable, sub-licensable, worldwide license to use content that you post. This license grant means that you have given Instagram the right to use any of your photos for free, for any reason, anywhere in the world. Instagram can also give those rights to a third party.

Under Instagram’s terms, you must ensure that you own the content that you post or at least have the right to grant Instagram these rights and licenses to the content. You are also responsible to pay any royalties, fees or money owed for the content that you post.

If someone is infringing your copyright on Instagram, you can report copyright abuse. Instagram may remove infringing content or disable the accounts of repeat copyright infringers. It is common practice to re-share content of other users of Instagram, if the source of the content is acknowledged.

For more on the do’s and don’ts of using Instagram, have a look at this post.