What is the correct structure of sodium thiosulfate?
Both of your Lewis structures are wrong.

In both of the shown structures, the solid line from sodium to oxygen/ sulfur implies that there is a discrete bond between these atoms. This is not the case. 

The sodium ions and the thiosulfate ions form ion pairs (or ion triples) and are attracted to each other via diffuse electrostatic interactions. Usually that is what we consider ionic bonding. 

In aqueous solution the ions will be very mobile. Sodium ions may coordinate to one or more thiosulfate ions, and therein to more than one site of the thiosulfate ion. At the same time thiosulfate ions may coordinate to more than just two sodium ion, or fewer. In such solutions, the solutes also coordinate to the solvent.
Ignoring any external field, the oxygen within the thiosulfate ion are equal.

The Lewis structure best describing the thiosulfate molecular ion is having a +2 formal charge at the central sulfur, and a -1 charge on each of the ligand atoms oxygen/ sulfur. (See below for details.)
One possible (and more accurate) Lewis structure is therefore:


(Note that the dashed lines do not represent bonds, but weakly coordinating interactions, but not necessarily all of them.)

More common are probably the following Lewis structure representations:



In solid state coordination of the ions depend on various factors, especially how much water is incorporated into the crystal structure. In any of these cases, it would be wrong to draw any bonds between the (molecular) ions. I'll include a few links to publications at the end of the post.



Additional thoughts of your input structures


  Which structure of sodium thiosulfate is correct? [...]
  


You noted correctly that in the first structure the central sulfur needs to "expand the octet". This would only be possible if d orbitals of said sulfur are involved. It has been proven many times that the contribution of d-orbitals in these hyper-coordinated species is minimal (usually below 1%).

The second structure is technically, or strictly speaking, not a Lewis structure, because dative bonds are not part of this description. However, if we allowed them as an extension, which has been proposed multiple times, then the arrow would need to point from the oxygen towards the central sulfur, as in those occurrences the more electronegative element would assume the higher electron density.



Disclaimer: The following section is a bit more advanced, and might not be appropriate to the casual reader, as another user pointed out that it might obfuscate the most important points about the structure of the anion.

Gas phase approach to determining the most likely Lewis structure

If we assume this molecule to be an ion-pair and calculate it in the gas phase, then we can draw a couple of conclusions, how a more accurate representation in the Lewis framework would look like.
I have performed a quick calculation on the DF-BP86/def2-SVP level of theory with Gaussian 09 and performed a natural bond order analysis (NBO 6.0) on it. The following picture shows the optimised geometry, alongside some inter-atomic distances.



As you can see, the sodium ions are coordinated to two oxygen, or one oxygen and a sulfur, respectively. The natural atomic charges reproduce well what I have initially stated, and reproduce well the Lewis structure I have suggested.

\begin{array}{rrr}
\text{Atom}&amp; \text{No}&amp; \text{Natural Charge}\\\hline
  \ce{S}  &amp; 1 &amp;   1.98 \\
  \ce{S}  &amp; 2 &amp;  -0.73 \\
  \ce{O}  &amp; 3 &amp;  -1.01 \\
  \ce{O}  &amp; 4 &amp;  -1.01 \\
  \ce{O}  &amp; 5 &amp;  -1.01 \\
  \ce{Na} &amp; 6 &amp;   0.91 \\
  \ce{Na} &amp; 7 &amp;   0.86 \\\hline
\end{array}

In terms of the NBO analysis, we can back this up, as it indicates no lone pair at the central sulfur (S1), three lone pairs at the oxygen (O3, O4, O5) and the ligand sulfur (S2), and four single bonds. There are vacant acceptor orbitals at the sodium ions (Na7, Na8). Here is an abridged version of the analysis:


     (Occupancy)   Bond orbital / Coefficients / Hybrids
 ------------------ Lewis ------------------------------------------------------
(skipping core orbitals)
  24. (1.98855) LP ( 1) S  2            s( 90.65%)p 0.10(  9.34%)d 0.00(  0.01%)
  25. (1.88728) LP ( 2) S  2            s(  0.00%)p 1.00( 99.89%)d 0.00(  0.11%)
  26. (1.84883) LP ( 3) S  2            s(  1.80%)p54.53( 98.09%)d 0.06(  0.11%)
  27. (1.97170) LP ( 1) O  3            s( 77.50%)p 0.29( 22.49%)d 0.00(  0.01%)
  28. (1.83558) LP ( 2) O  3            s(  0.67%)p99.99( 99.25%)d 0.11(  0.07%)
  29. (1.82232) LP ( 3) O  3            s(  0.03%)p99.99( 99.89%)d 2.42(  0.08%)
  30. (1.97170) LP ( 1) O  4            s( 77.50%)p 0.29( 22.49%)d 0.00(  0.01%)
  31. (1.83558) LP ( 2) O  4            s(  0.67%)p99.99( 99.25%)d 0.11(  0.07%)
  32. (1.82232) LP ( 3) O  4            s(  0.03%)p99.99( 99.89%)d 2.42(  0.08%)
  33. (1.96648) LP ( 1) O  5            s( 77.67%)p 0.29( 22.33%)d 0.00(  0.01%)
  34. (1.83753) LP ( 2) O  5            s(  0.02%)p99.99( 99.91%)d 4.78(  0.07%)
  35. (1.82147) LP ( 3) O  5            s(  0.00%)p 1.00( 99.93%)d 0.00(  0.07%)
  36. (1.95903) BD ( 1) S  1- S  2
               ( 56.47%)   0.7515* S  1 s( 25.58%)p 2.86( 73.06%)d 0.05(  1.36%)
               ( 43.53%)   0.6598* S  2 s(  7.70%)p11.86( 91.35%)d 0.12(  0.95%)
  37. (1.97694) BD ( 1) S  1- O  3
               ( 34.82%)   0.5901* S  1 s( 24.89%)p 2.97( 74.04%)d 0.04(  1.07%)
               ( 65.18%)   0.8073* O  3 s( 21.84%)p 3.57( 77.98%)d 0.01(  0.19%)
  38. (1.97694) BD ( 1) S  1- O  4
               ( 34.82%)   0.5901* S  1 s( 24.89%)p 2.97( 74.04%)d 0.04(  1.07%)
               ( 65.18%)   0.8073* O  4 s( 21.84%)p 3.57( 77.98%)d 0.01(  0.19%)
  39. (1.97758) BD ( 1) S  1- O  5
               ( 34.64%)   0.5886* S  1 s( 25.28%)p 2.91( 73.66%)d 0.04(  1.06%)
               ( 65.36%)   0.8085* O  5 s( 22.34%)p 3.47( 77.48%)d 0.01(  0.18%)
 ---------------- non-Lewis ----------------------------------------------------
  40. (0.05958) LV ( 1)Na  6            s( 95.91%)p 0.04(  3.91%)d 0.00(  0.18%)
  41. (0.10019) LV ( 1)Na  7            s( 96.97%)p 0.03(  2.93%)d 0.00(  0.10%)
  42. (0.25853) BD*( 1) S  1- S  2
               ( 43.53%)   0.6598* S  1 s( 25.58%)p 2.86( 73.06%)d 0.05(  1.36%)
               ( 56.47%)  -0.7515* S  2 s(  7.70%)p11.86( 91.35%)d 0.12(  0.95%)
  43. (0.24584) BD*( 1) S  1- O  3
               ( 65.18%)   0.8073* S  1 s( 24.89%)p 2.97( 74.04%)d 0.04(  1.07%)
               ( 34.82%)  -0.5901* O  3 s( 21.84%)p 3.57( 77.98%)d 0.01(  0.19%)
  44. (0.24584) BD*( 1) S  1- O  4
               ( 65.18%)   0.8073* S  1 s( 24.89%)p 2.97( 74.04%)d 0.04(  1.07%)
               ( 34.82%)  -0.5901* O  4 s( 21.84%)p 3.57( 77.98%)d 0.01(  0.19%)
  45. (0.24610) BD*( 1) S  1- O  5
               ( 65.36%)   0.8085* S  1 s( 25.28%)p 2.91( 73.66%)d 0.04(  1.06%)
               ( 34.64%)  -0.5886* O  5 s( 22.34%)p 3.47( 77.48%)d 0.01(  0.18%)

Key: LP = lone pair, BD = bond, LV = lone valency, BD* = "antibond"


Alternative coordinations

There are at least two other structures of the stoichiometry $\ce{Na2S2O3}$,  essentially with the same bonding motif, but different coordinations of the sodium atoms. The first is about $\pu{2.4 kJ mol-1}$ more stable than the previously shown.



The second is about $\pu{3.2 kJ mol-1}$ higher in energy.



These are electronic energies only, but they show that the sodium atoms are not discretely bound and may interchange positions quite freely.
Given that disodium thiosulfate probably only exists in a crystal lattice which cannot be described by Lewis structures, or in solution where we have to worry about other effects and equilibria, it is obvious, that there cannot be a single Lewis structure for this compound. It simply does not exist as a discrete molecule.



Literature (selection) of crystal structures


The crystal structure of anhydrous sodium thiosulphate. Sandor, E.; Csordas, L.; Acta Cryst. 1961, 14, 237-243. DOI: 10.1107/S0365110X61000802 Crystallography online: Crystal Structure of Na2 (S2 O3)
On Polymorphism of $\ce{Na2S2O3}$. von Benda, H.; von Benda, K.; Z. Naturforsch. B 1979, 34, 957-968. DOI:
10.1515/znb-1979-0716 Alternative direct link (pdf) Crystallography online: Crystal Structure of Na2 (S2 O3), Crystal Structure of Na2 (S2 O3), Crystal Structure of Na2 (S2 O3)
Rerefinement of sodium thio­sulfate penta­hydrate. Prasad, S. M.; Rani, A.; Acta Cryst. E, 2001, 57, i67-i69. DOI: 10.1107/S1600536801011175 Crystallography online: Crystal Structure of Sodium thiosulfate pentahydrate
Structure of a new sodium thiosulfate hydrate. Hesse, W.; Leutner, B.; Böhn, K.-H.; Walker, N. P. C.; Acta Cryst. C 1993, 49, 363-365. DOI: 10.1107/S0108270192009399 Crystallography online: Crystal Structure of sodium thiosulphate 2/3-hydrate
The crystal structure of sodium thiosulphate pentahydrate. Taylor, P. G.; Beevers, C. A.; Acta Cryst. 1952, 5, 341-344. DOI: 10.1107/S0365110X52001027