You are listed on the 1099-SA because you are the owner of the HSA. There is no such thing as a joint HSA, only the Family Plan for the HDHP that allows coverage of others in the family (but see below).
Therefore, only you can report the contributions and distributions.
However, as for distributions, I need to check something. Is your domestic partnership considered a legal marriage in your state?
I ask because you can make payments for qualified payments only to the people listed below.
"Qualified medical expenses are those incurred by the following persons.
1. You and your spouse.
2. All dependents you claim on your tax return.
3. Any person you could have claimed as a dependent on your return except that:
a. The person filed a joint return,
b. The person had gross income of $4,050 or more, or
c. You, or your spouse if filing jointly, could be claimed as a dependent on someone else's 2016 return." - IRS Pub 969
Who is a spouse? If you are a same-sex couple, the IRS says this in Pub 969:
"Federal tax benefits for same-sex married couples. For federal tax purposes, marriages of couples of the same sex are treated the same as marriages of couples of the opposite sex. The term "spouse" includes an individual married to a person of the same sex. However, individuals who have entered into a registered domestic partnership, civil union, or other similar relationship that isn't considered a marriage under state law aren't considered married for federal tax purposes."
If you are a heterosexual couple, then you also have to look to your state law to see if this domestic partnership is the same as marriage.
If you two are not legally married and your other half can't be construed as a dependent (except as noted above), then you can't spend your HSA money on your other half. That is, just because your company allows you to extend benefits to your partner, this does not mean that he/she is a "spouse" in federal terms.
You will want to consult with someone experienced in tax law in your state to see where you stand.