Anti-Mage is a strong carry and Shadow Shaman a strong disabler/pusher. I assume the two of them are laning together, but if not the same will still somewhat apply. Trying to outcarry Anti-Mage usually isn't a very good prospect. If you're really determined to try to do that, Riki and Slardar are your best options.
Otherwise, you need to have someone strong to harass Anti-Mage when he's at his weakest and slow down his farm. If he's allowed to farm easily for 10-15 minutes he's incredibly harder to deal with in teamfights. You can't expect to kill him, however: blink is too good of an ability. Instead, you want someone who's not going to be threatened by him much in lane.
Shadow Shaman is not a very good babysitter because he needs early levels. His autoattack range, damage, and armor are all low, and so he is easily outclassed by an enemy harasser. Once Shadow Shaman is more subdued, you can turn to keeping the Anti-Mage from getting easy farm.
I recommend Lich or Venomancer for this purpose, but you'll need a lane partner with a stun as well to prevent Shadow Shaman from just gripping you. Anyone with a stun will do, but I'm not sure if the bots will be smart enough to use their stun as an interrupt. (At this point I should mention that you guys should be laning against each other if you're going 2v2 with 3 bots per team, otherwise you're just seeing who's better at taking advantage of dumb bots.) Other heroes like Shadow Demon, Obsidian Destroyer, and Bane can be useful counters to Anti-Mage at various stages of the game because of their spells, while heroes like Leshrac and Death Prophet can push towers and make it more difficult for Anti-Mage to farm.
Countering Rhasta (Shadow Shaman) is a lot more simple. You just need to be coordinated to quickly stop his Shackle, and don't be out of position enough to be Hexed. If you get ward-trapped, either use Phase Boots to walk out or kill one of the edge (not corner) wards and then walk free.
Ghost Scepter and Force Staff deserve special notice as great item pickups against both of these heroes. Shadow Shaman's wards won't be a threat and Anti-Mage can't stay on you (but be wary of his allowing his ultimate to do extra damage with Ghost Scepter). Force Staff also helps save teammates (Force Staff them away from the Anti-Mage, forcing him to commit more heavily for a kill attempt).
tl;dr: Lane Lich and a stunner against the AM. Get pushing power elsewhere. Deny him farm, use Frost Armor/Force Staff/Ghost Scepter in midgame to prevent him from killing people quickly with melee attacks. Push heavily during this time to earn a gold lead and hopefully take a set of Barracks quickly. You don't want to go lategame against an Anti-Mage without a similarly powerful carry like Riki/Slardar/Faceless Void.
Foreword: I am focusing on DotA 2 since i stopped playing DotA 1 for quite a while now.
Offensive
Stuns will hold Morphling in place but wont interfere with his morph ability in any way.
Even Void's Chronosphere cannot stop his health gain.
Therefore you need a silence whether its an ability or an item to counter Morphling.
Important sidenote: Mantastyle does remove Orchid silence.
Naix with Orchid can be good because even if Morphling morphed into strength Naixwill deal more damage.
Clinkz with Orchid works well too because of his burst damage.
Silencer is very good because his Ultimate is not a target spell and global giving your team the opportunity to counter Morphling anytime.
(thanks to Decency for mentioning those 2 in his answer):
Ancient Apparation is another great pick vs Morphling because it prevents all kinds of healing including those from skills or items.
Lion's Mana Drain and Hex can destroy an illusion immediately. Combined with his long disable overall and his nuke he can be a threat at all times especially with Blink Dagger.
Defensive
To protect yourself against the enormous nuke of an Ethereal Blade-Morphling you should buy magic armor and a Mekansm. (Chen's Hand of God can be a substitute for Mekansm. But since you need to heal yourself quickly i would not rely on another players reaction. Having both is even better)
Furthermore BKB and spells like Repel help a lot.
Since EB-Morphling needs as much agility as possible to nuke hard he wont have much HP while nuking. This opens a window for a Blademail that can hit Morphling hard and enables the possibility for a good trade (your support for their midgame-carry)
Since Linkens Sphere is annoying you should pick heroes who can afford wasting a spell to disable his Linkens and have another good stun as a follow up.
For example: Bane, Lion, Shadow Shaman, Shadow Demon, Ogre, Rubik, Disruptor.
Best Answer
I would put the strongest player mid, as the outcome of the midlane usually dictates the mid-game. The mid-game for me is after the laning phase, when people start going for objectives (such as towers, roshan or the bounty rune). If your team wins the midlane, that means that the enemy mid-player is forced out of his lane, which gives you/your team access to their mid tower and the two bounties. You 'secure' the river in a way, and this is also where you have access to roshan, the (bounty) runes, and potential easier ganks to the sidelanes. Finally, most played mid heroes scale very well into this midgame, and having a good player have more gold and levels over the other midplayer makes them a huge threat to the side lanes (where the carries are).
I would put a significantly lower skilled team member on a support. In the current meta (7.19, duo-lanes) where there is a scaling/carry hero and a support hero in both sidelanes for each team, I would put this team member on the offlane support. In the case of a trilane, I would let him play the second support. What I mean with that is that they should buy more expensive support items (Force Staff, Urn of Shadows, Glimmer cape) and occasionally wards, and let my equally skilled team member be the full support that keeps on warding etc. This way, you take away the pressure of have everything fall on your less skilled team member to help your main carry in the safelane, and just let him play his hero and focus on that. I'd suggest a hero with a stun, as they have a reliable and easy way to help out the carry in that lane. Finally, you can have your other (good) support help out your less skilled team member by suggesting ward placements or organizing ganks.
I would not let my less skilled team member play either carry or mid, as these require to take advantage of space to farm and get more gold and then utilize that gold well to buy the correct items for the situation and this is something that less skilled players seem to lack in most of the time.