Answer: $1, 908, 1011, 1918, 2021$ is the only possibility.

Let $(a, 1919 - a)$ and $(b, 1919 - b)$ be the two pairs of numbers with sum $1919$. If these sums had a common addend then both addends would be the same whence the choices of two numbers would not be different. Thus $a, b, 1919 - a$ and $1919 - b$ are pairwise distinct. Let $x$ be the fifth chosen number; then the total sum of chosen numbers is $3838 + x$.

Similarly, we see that the five numbers must be representable as $c, 2929 - c, d, 2929 - d, y$; their total sum is $5858 + y$. From $3838 + x = 5858 + y$, one gets $x - y = 2020$. The only possibility for satisfying the latter equality is $x = 2021, y = 1$. Hence one of $a, b, 1919 - a, 1919 - b$ equals $1$ and one of $c, d, 2929 - c, 2929 - d$ is $2021$; w.l.o.g., $a = 1$ and $c = 2021$. Then the set of chosen numbers contains also $1919 - 1 = 1918$ and $2929 - 2021 = 908$. W.l.o.g., $b = 908$. The last chosen number must be $1919 - 908 = 1011$.

Let the chosen numbers be $a, b, c, d, e$. W.l.o.g., $a + b = 1919$. If the other way to obtain $1919$ as the sum of two chosen numbers used either $a$ or $b$ as addend then the other addend would have to be $b$ or $a$, respectively, which is not allowed. Hence, w.l.o.g., $c + d = 1919$.

Similarly, one must use four distinct numbers to get $2929$ as the sum of two chosen numbers in two different ways. If these four numbers were $a, b, c, d$, then $a + b = c + d = 1919$ implies $a + b + c + d = 3838$ while also $a + b + c + d = 2929 + 2929 = 5858$. The contradiction shows that one must use the number $e$ in at least one pair of numbers with sum $2929$. W.l.o.g., $d + e = 2929$. The other pair of numbers with sum $2929$ must use exactly two numbers among $a, b, c$, but $a + b = 1919$. Hence, w.l.o.g., $b + c = 2929$.

These equalities together imply $b = 1919 - a$, $c = 2929 - b = 1010 + a$, $d = 1919 - c = 909 - a$ and $e = 2929 - d = 2020 + a$. The latter implies $a = 1$ and $e = 2021$. Substituting $a = 1$ into the other equalities gives $b = 1918$, $c = 1011$ and $d = 908$.