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For some reason it has been very difficult to find the right answer for this question as there's so much conflicting information to be found online. Some mention a reaction where carbonic acid is made, others mention carbonates and hydroxide end product but nothing seems to be precise.
But can the reaction be as simple as this (a double displacement reaction) where magnesium bicarbonate and sodium chloride are the end products: $$ \ce{2NaHCO3 + MgCl2 -> 2NaCl + Mg(HCO3)2} $$
Having actually perform this reaction, I can say with certainty that when you mix aqueous sodium bicarbonate with aqueous magnesium chloride, there is no apparent reaction. This is consistent with an equilibrium reaction (actually, just ions):
$$\ce{2 NaHCO3 + MgCl2 <=> 2 NaCl + Mg(HCO3)2}$$
However, upon brief heating in a microwave (for example), a white precipitate of magnesium carbonate develops indicating the reaction has moved to the right.
$$\ce{2 NaHCO3 + MgCl2 ->[Heat] 2 NaCl + MgCO3(s) + CO2 (g) + H2O}$$
The chemicals must be in water, since magnesium bicarbonate decomposes when dried. So I will assume that the mixing is done in water. To start with, both salts are strong electrolytes and so they immediately dissociate to give $\ce{Na+}$ ions, $\ce{HCO3-}$ ions, $\ce{Mg^{2+}}$ ions, and $\ce{Cl-}$ ions. As Brenton Home has pointed out, none of the possible compounds formed from these are insoluble and there is no gas formation due to the decomposition of the bicarbonate because the resulting solution is not acidic.
So the answer is that there is no reaction.
I thought I would start out in saying that I think this is a random, but great question. This is because as you said there's so little information out there regarding this sort of reaction so it was an interesting little puzzle for me to figure out for you.
My guess would be that the reaction you detailed would not occur. This double replacement reaction would occur in aqueous solution as otherwise the product of magnesium bicarbonate probably wouldn't have a chance of forming since at least one other alkali earth metal bicarbonate salt, calcium bicarbonate, does not exist outside of aqueous solution. I tell you this piece of information instead of whether magnesium bicarbonate exists outside of aqueous solution because I just don't know for certain whether it does. Evidence for this theory that it doesn't exist out of aqueous solution includes (in addition to the point raised above about calcium bicarbonate):
Since it occurs in aqueous the only driving force to the reaction would be the production of an insoluble salt, in your reaction, however, there is no insoluble products that results.
Therefore I suspect an alternate reaction would occur. Namely:
$$\ce{2NaHCO_3 + MgCl_2 -> 2NaCl + MgO + H_2 O + 2 CO_2}$$
As this reaction would lead to three insoluble products and thus would invoke a greater drive for the reaction.
There can be a reaction, maybe. A simple double displacement gives nothing, but perhaps magnesium bicarbonate can "disproportionate" in an acid base sense:
$\ce{Mg^2+ + 2HCO3- -> MgCO3(s) + CO2 + H2O}$
where the reaction would be driven by the magnesium carbonate precipitating. I would need to look up the relevant equilibrium constants (acid-base for carbonate amd bicarbonate ions, precipitation of the magnesium carbonate) and be given solution concentrations to see whether we really get this reaction.
