Local storage with full admin rights. Much faster, more reliable, no problems with file locking mechanisms.
Editing media over network can be done. However the media bandwidth requirements have to be met for playback and you will have more latency in realtime playback and rendering. Gbit networks are normally not enough for many of the higher resolution capable codecs. 10gbe will likely be required at some point if not now.
How many users are you talking? What sort of media are you working with? And are you working on Macs or PCs?
Despite the scary sounding Adobe official cautions network storage, whether it be NAS or SAN, generally work fine. There's two big factors to consider when looking at an appropriate network storage device: 1) what your throughput demands are, and 2) what the network storage device can pump out.
Regarding the first the bandwidth requirements for compressed HD or 4k, like MPEG4 or AVCHD footage, are pretty minimal (around 5 - 10 MB/s) so an off the shelf NAS would be fine. For more heavy duty content, like HD Prores 4444 or uncompressed footage, you'll start to push the gigabit connection particularly if multiple users are accessing the storage or your timeline references multiple video tracks. A 4k uncompressed stream is about 480 MB/s for example!
On the second point a rule of thumb is the more drives a storage device has the faster it will go. That's particularly relevant if we're talking spindle drives, and less so if we're talking SSD's because they just GO regardless of the number. Speed will be further impacted by the configuration of the array. You can get a rough (VERY rough) handle on array speed here: RAID Performace Calculator - WintelGuy.com -. That's one half of it. The second is the device interface. A 1GbE network cable can feed about 115 MB/s. That is rarely sustained, so there are peaks and troughs to the speed. If you deal with uncompressed footage you'll saturate that pretty quickly, so a jump up to a device with a 10GbE card and a corresponding 10GbE card or adapter for your computer may be required. You'll also need a 10GbE switch unless you're going direct device to device. A 10GbE interface theoretically peaks at 1250 MB/s, so the bottleneck in this case will be the array drive speeds. Our 12 bay NAS for example, can only manage about 400 - 600 MB/s. But even still you'll get a nice bump up from a 1GbE NAS.
Work out A, your throughput requirement, and find B, a product that meets these requirements, and you're 99% there. For ease of management and cost effectiveness I'd personally recommend a Synology or QNAP NAS. These come in 1GbE and 10GbE flavours.
My post at the bottom of this thread is how we setup our Premiere projects to work on a NAS:
All up, a NAS or SAN solution is going to be a lot more expensive than a direct attached storage device and a little more fiddly to manage. Great for multiple users, not so great to jump through a specific IT requirement hoop.
Uncompressed HD bit rate is anywhere from 200MB/s to 500MB/s depending on frame rate. 4K uncompressed is far higher than 480MB/s and can push over 1.2GB/s with DPX files.