Do I need to own both the Dungeon Master's Guide & the Monster Manual in order to run Curse of Strahd?
I currently only have the Player's Handbook and the starter pack.
Large parts of the DMG and MM are now available via the open Systems Reference Document (SRD), which is a free to download 400+ page PDF. There are some sites that have organized this document into a Wiki format which will make it even easier to find the stuff you need.
For example, the Curse of Strahd features a special magic weapon. That weapon is not in the book, but is available in the SRD.
As an extra note, digital versions of the maps are available for purchase from the artist, Mike Schley. These maps include "player-friendly" and "grid-free" versions which are a step beyond just "scanning the book". There are also some free B&W maps of the region available for free.
Also note that the a quest covering the first 3 levels of CoS was released for free by Wizards. The Death House can be downloaded as a 12-page PDF. Again, larger maps are available from the source above.
UPDATE: I manually walked through the book to double-check for magic items not in the SRD. Here's a list of what I found:
My copy finally arrived after I ordered it online, and on page 5 of the book, under Running the Adventure it states:
It assumes you have the fifth edition of the Player's Handbook, Dungeon Master's Guide, and Monster Manual
The module has some of the statistics for monsters in the back, but only ones specific to the module (ie Strahd himself, special creatures found only in Barovia). Some other statblocks (like Animated Armor, Shambling Mound, etc), you will either need the Monster Manual for or you will need to get the statblocks elsewhere ahead of time.
Additionally, some of the items such as the (spoiler)
do not have full statistics either, so the DMG would be needed unless you houserule or look them up elsewhere.
Rules for playing the game can be found in the basic set or the PHB, and will probably suffice.
No. Adventurer's League, the official Wizards of the Coast backed organized play league, requires only that you have a copy of the basic rules and an adventure. If the most formal version of D&D play doesn't require more than that you should be fine running without them in less-formal settings as well. That said, the Dungeon Master's Guide and Monster Manual are both very nice things to have even if they aren't necessary.