Hooboy. This may be prohibitive to bring to full code, so let me discuss some options to get the most safety increase for the least buck.
Romex NM-B is not legal for direct-burial or for running in conduit. But what really has me spooked is bootlegging a neutral off that ground wire. If anything breaks in that ground wire, it will energize every ground in your electrical system at 120VAC - touching a conduit, the panel, light switch screws, all the things that are supposed to be safe will be dangerous! Far away from the house you could be dead before they find you! Do not leave this situation to fester merely because bringing it all the way to current code would be hard: Don't let "perfect" be the enemy of the "good". Every option here restores the proper function of the safety ground and separates neutral from it.
Ed Beal discusses an issue with the ground rod.
I am willing to assume some defects may be grandfathered or permitted via local exemption.
Option 1: Go 120V-only
If you can bear having only 120V, then remove all markings from the white wire. In the main panel, black goes to a 120V breaker (or one side of a 240V breaker), white to neutral bar, and bare to ground bar (they may be the same in the main panel). In the sub-panel, black wire goes to one phase, white to the neutral bar, bare to the ground bar which are isolated. Every other row of breakers will not work - just move your needed breakers to rows that do work.
Cost: $30ish per building, for a separate neutral bar for the sub-panel
Option 2: Go 240V-only - temporary transformers for 120V
If your primary loads are 240V-only and/or multi-voltage loads (lighting, some gadgets), and your 120V loads can be hooked up temporarily as needed, easy - just reconfigure for 240V-only. Stop using neutral; the bare wire is ground-only.
Mark the white wire with tape on both ends to designate it a "hot". In the main panel, black and remarked-white to a 240V breaker. Bare to the ground bus. In the sub-panel, black and remarked-white to opposite "hots", bare to the ground bar. No neutral bar - no neutrals will exist in this panel! No exceptions! Every white wire in wiring will be marked with tape as it's not a neutral.
To power 240V loads, done. Many machines will run on 120-240V. Lighting particularly - check your fluorescent lights, the newest ballasts are 120-277V automatic switching, if yours isn't, just change ballast, they're $10-20. Good time to upgrade to T8 bulbs while you're at it. Many LED bulbs are 120-277V, I've been replacing 175W barn lights with 15W LED lights that are $10 and multivoltage. Old mercury, sodium and halide lights have transformers, and they are often jumpable for 240V. For plug-connected devices that can accept 240V, change your wired receptacles and plugs to NEMA 6-15 or 6-20.
To temporarily power loads that are 120V, use a common step-up/step-down transformer.
Cost: est. $250 per building for new bulbs, ballasts, outlets, plugs, power supplies and a step-up/down transformer.
Option 3: Go 240V, install a 120V transformer permanently
This is like Option 2, except with a permanent 120V transformer feeding a third panel.
I often find 5KVA transformers around "used" for about $100. These have 240V on the primary, and 20 amps of 120/240V split-phase on the secondary (jumpable for 40 amps of 120V only). Do the sub-panel per option 2, and add a 20A breaker to feed this transformer. The third panel gets its hots and neutral from the secondary of the transformer. It gets its ground from the ground rods feeding the sub-panel. Because the third panel is fed from a transformer, it's actually not a sub-panel at all - it's a main panel, and you do bond neutral to ground here. The transformer makes this safe.
You only have 20A per leg (or 40A at 120V-only) - so keep the loads here to a minimum.
Cost: est. $300 per building for used transformer, enclosure, additional mini-panel, breakers and wiring.
Option 4: Use a transformer for all loads.
Here, we use a larger transformer (12 KVA or larger, single phase) to feed the entire sub-panel. Like above, this makes it a main panel.
In the house's main panel, we mark the white wire with tape, and punch it down into a 240V breaker. Ground goes to the ground bar. At the transformer, we jumper the transformer's primary for 240V and attach the black and marked-white wire. Ground goes to the transformer chassis or enclosure.
We jumper the transformer's secondary for 120/240 split-phase. Its hots and neutral go to the 2 hots and neutral bar. Being fed by a transformer makes it a main panel, so ground and neutral are bonded together and the ground must go to the rods and also to the transformer chassis. The now-main panel is hooked up in the normal way. This gives you best of all worlds at higher cost.
Cost: est. $500 per building for a used transformer and wiring; $1000 for a new one.
Option 5: Pull 4 wires (in existing conduit??)
ThreePhaseEel addresses this well in his answer. You're pushing the limits of what that conduit can handle, out and in, and it's going to be one miserable pull. I prefer to install conduit much larger than the wires need, mainly to make the pull easier (and provide room for expansion). I would bring in an electrician for this - simply because they'll have all the right pulling tools on the truck, and the skill to do the pull without tearing up the wire. The right tools make all the difference in the world.
Keep in mind if there's any damage to this conduit, the pull will fail and you will end up digging up the yard. Water in the conduit is no big deal; that's why you use THWN wire.
And don't presume it's in conduit. If it's direct-burial cable, Code requires the cable exit the ground in conduit. That would explain using barely-big-enough conduit that he couldn't possibly pull 200 feet through. If it's direct-buried, you'll have to re-trench it.
Cost: est. $700-1200 per building: $400 for wire; $300 for tools or $800 to hire an electrician for the pull. If the conduit is not pullable, more.
Best Answer
Conduit is your friend here
As a general rule, using direct bury cable for an outdoor feeder is penny-wise and pound-foolish due to the cost of digging it up and replacing it when an upgrade is desired (a trencher rental + labor) compared to the cost of installing conduit (Schedule 80 PVC, or sometimes rigid metal conduit even) to begin with. A 2" Schedule 80 PVC conduit between the buildings provides sufficient space at 742mm2 for 3 4/0 Al XHHW-2 conductors (at about 525mm2 total) and a 6AWG bare copper ground (which adds another 13mm2, to make for a total of 538mm2), while if you really want to go for broke, you could use 3" for easy pulling and extra expansion room (not that you'd need more than about 100-125A to an outbuilding, that is).
If you can't afford to do anything but stick with the cable...
If your only option is sticking with the cable you have, though, then what you can do is wrap a ground wire in with the cable as you lay it, as the NEC 300.3(B) requirements allow multiple individual conductors or cables in the same trench to be used as part of the same circuit:
In particular, you'll need a 6 AWG bare copper ground wire for this, instead of the 8 AWG normally called for by a 100A circuit, due to the 250.122(B) size increase (which works out to a a size increase of about 1.6x over the base size from the table):
You will need separate ground and neutral bars in the subpanel, no matter what
As a result of running a separate ground wire back to the main panel, you will need to have separate ground and neutral bars in the subpanel, with all the ground wires landing on the ground bars and all the neutral wires landing on the neutral bars. Note that unless your panel shipped with extra ground bars factory installed, you will need to go to a local electrical supply house and buy a set that is appropriate for the make of your panel, or order the correct manufacturer's part online.
Furthermore, you will need to pull the (green) bonding screw or strap out of the subpanel and toss it, as it will not be needed -- neutral only meets ground at the main panel, as long as a separately derived systems is not sticking its nose into the tent, which only happens if you have a distribution transformer somewhere downstream of the main panel, or with certain generator configurations.
And you will also need that ground stake at the shed, as well
You will need some more of that 6AWG bare copper to connect the ground stakes (two of them, 8' deep and 8' apart) at the shed to the ground bar at the shed panel. This provides a path for stray natural electricity to get back to the earth, while the ground wire between the main panel and the subpanel provides a path for wayward utility electricity to get back to the utility.