To avoid such straightforward examples, let us assume that $G$ is a non-abelian finite simple group and $\rho_i \neq 1 $. Surprisingly, such phenomenon is still possible in this case, but quite rare. It occurs just on $7$ (among $114$) such groups of order less than $2\times 10^7$, namely, $\PSp(4,3)$, $M_{12}$, $A_9$, $\PSp(6,2)$, $M_{23}$, $\PSU(5,2)$, $2F(4,2)'$. See the computation in Appendix.

More specifically, for the alternating groups $A_n$, with $5 \le n \le 20$, this phenomenon occurs just for $n=9,16$, with $(\dim(V_1),\dim(V_2))=(8,21), (15,12012)$, respectively.

Question: For which non-abelian finite simple groups this phenomenon occurs? more specifically, for which alternating groups $A_n$? if and only if $n$ is a square? and $\dim(V_1) = n-1$?

We know that we can avoid $\PSL(2,q)$, see here.

gap> it:=SimpleGroupsIterator(60,20000000);; for g in it do N:=Name(g);; if Size(N)<6 or List([1..6],i->N[i]) <> "PSL(2," then Print(Name(g),"\n",Order(g),"\n");; Phenomenon(g);; fi; od;
...
PSp(4,3), M12, A9, PSp(6,2), M23, PSU(5,2), 2F(4,2)'

Computation for the alternating group $A_n$, for $5 \le n \le 20$:

gap> for n in [5..20] do Print(n,"\n");; g:=AlternatingGroup(n);; Phenomenon(g);; od;
...
A9, A16

Script

Phenomenon:=function(g)
    local irr,r,L,i,j,k;
    irr:=Irr(g);
    r:=Size(irr);
    for i in [2..r] do
        for j in [i..r] do
            L:=List([1..r],k->ScalarProduct(irr[i]*irr[j],irr[k]));;
            if Number(L,i-> i=0)=r-1 and Number(L,i-> i=1)=1 then 
                Print([i,j],"\n");; 
            fi;
        od;
    od;
end;;

To avoid such straightforward examples, let us assume that $G$ is a non-abelian finite simple group and $\rho_i \neq 1 $. Surprisingly, such phenomenon is still possible in this case, but quite rare. 

It occurs just on $10$ (among $204$) such groups of order less than $2.7\times 10^8$ (see the computations in Appendix), namely,
$$\PSp(4,3), \ M_{12}, \ A_9, \ \PSp(6,2), \ M_{23}, \ \PSU(5,2), \ 2F(4,2)', \ O_+(8,2), \ ^3D(4,2), \ M_{24}.$$

More specifically, for the alternating groups $A_n$, with $5 \le n \le 21$, this phenomenon occurs just for $n=9,16$, with $(\dim(V_1),\dim(V_2))=(8,21), (15,12012)$, respectively.

For the sporadic groups, it occurs for $9$ ones (among $26$), namely,
$$M_{12}, \ M_{23}, \ M_{24}, \ Co_3, \ Co_2, \ Th, \ Co_1, \ F_{3+}, \ M.$$ Warning: This last list differs from (the complement of) the list in this review mentioned by Nick Gill in this comment.

Question: For which non-abelian finite simple groups this phenomenon occurs? more specifically, for which alternating groups $A_n$? if and only if $n$ is a square? and $\dim(V_1) = n-1$?

For general non-abelian finite simple groups (we know that we can avoid $\PSL(2,q)$, see here):

gap> simpnames:= AllCharacterTableNames( IsSimple, true, IsAbelian, false, IsDuplicateTable, false : OrderedBy:= Size);
gap> for nam in simpnames do if Size(nam)<3 or List([1..3],i->nam[i]) <> "L2(" then ct:=CharacterTable(nam); Print(nam," ",Size(ct)," ",c,"\n"); Phenomenon(ct); fi; od; 
...
U4(2), M12, A9, S6(2), M23, U5(2), 2F4(2)', O8+(2), 3D4(2), M24, ... the list should be incomplete after, because AllCharacterTableNames contains finitely many simple groups

For the alternating group $A_n$, for $5 \le n \le 21$:

gap> for n in [5..21] do Print(n,"\n");; ct:=CharacterTable("Alternating", n); Phenomenon(ct);; od;
...
A9, A16

For the sporadic groups:

gap> spornames:= AllCharacterTableNames(IsSporadicSimple, true, IsDuplicateTable, false : OrderedBy:= Size);;
gap> for nam in spornames do ct:=CharacterTable(nam); Print(nam," ",Size(ct),"\n"); Phenomenon(ct); od;    
M12, M23, M24, Co3, Co2, Th, Co1, F3+, M

Script

LoadPackage("ctbllib"); LoadPackage("atlasrep");
Phenomenon:=function(ct)
    local irr,r,L,i,j,k;
    irr:=Irr(ct);
    r:=Size(irr);
    for i in [2..r] do
        for j in [i..r] do
            if IsIrreducible(irr[i]*irr[j]) then
                Print([i,j],"\n");; 
            fi;
        od;
    od;
end;;

• If $V_1$ or $V_2$ is one-dimensional, then $ \rho_1 \otimes \rho_2 $ is irreducible.

• Let $G$ be the product $ G_1 \times G_2 $. If $ \rho_i $ is an irreducible representation of $ G_i $, then the representation $\rho_1 \otimes \rho_2 = (\rho_1 \otimes 1) \otimes (1 \otimes \rho_2)$ is irreducible for $ G_1 \times G_2 $.

To avoid such straightforward examples, let us assume that $G$ is a non-abelian finite simple group and $\rho_i \neq 1 $. Surprisingly, such phenomenon is still possible in this case, but quite rare. It occurs just on seven such groups of order less than $2\times 10^7$, namely, $\PSp(4,3)$, $M_{12}$, $A_9$, $\PSp(6,2)$, $M_{23}$, $\PSU(5,2)$, $2F(4,2)'$. See the computation in Appendix.

it:=SimpleGroupsIterator(60,20000000);; for g in it do N:=Name(g);; if Size(N)<6 or List([1..6],i->N[i]) <> "PSL(2," then Print(Name(g),"\n",Order(g),"\n");; Phenomenon(g);; fi; od;

PSp(4,3), M12, A9, PSp(6,2), M23, PSU(5,2), 2F(4,2)'

for n in [5..20] do Print(n,"\n");; g:=AlternatingGroup(n);; Phenomenon(g);; od;

A9, A16

• If $V_1$ or $V_2$ is one-dimensional, then $ \rho_1 \otimes \rho_2 $ is irreducible.

• Let $G$ be the product $ G_1 \times G_2 $. If $ \rho_i $ is an irreducible representation of $ G_i $, then the representation $\rho_1 \boxtimes \rho_2 = (\rho_1 \boxtimes 1) \otimes (1 \boxtimes \rho_2)$ is irreducible for $ G_1 \times G_2 $.

To avoid such straightforward examples, let us assume that $G$ is a non-abelian finite simple group and $\rho_i \neq 1 $. Surprisingly, such phenomenon is still possible in this case, but quite rare. It occurs just on $7$ (among $114$) such groups of order less than $2\times 10^7$, namely, $\PSp(4,3)$, $M_{12}$, $A_9$, $\PSp(6,2)$, $M_{23}$, $\PSU(5,2)$, $2F(4,2)'$. See the computation in Appendix.

gap> it:=SimpleGroupsIterator(60,20000000);; for g in it do N:=Name(g);; if Size(N)<6 or List([1..6],i->N[i]) <> "PSL(2," then Print(Name(g),"\n",Order(g),"\n");; Phenomenon(g);; fi; od;
...
PSp(4,3), M12, A9, PSp(6,2), M23, PSU(5,2), 2F(4,2)'

gap> for n in [5..20] do Print(n,"\n");; g:=AlternatingGroup(n);; Phenomenon(g);; od;
...
A9, A16

Background:

A representation $ \rho: G \to GL(V) $ of a group $ G $ on a (complex) vector space $ V $ is said to be irreducible if there are no nontrivial invariant subspaces under the action of $ G $. A tensor product of two irreducible representations $ \rho_i: G \to GL(V_i) $, $i=1,2$, is a representation on the tensor product space $ V_1 \otimes V_2 $, defined by $ \rho_1(g) \otimes \rho_2(g) $ for $ g \in G $.

To avoid such straightforward examples, let us assume that $G$ is a non-abelian finite simple group and $\rho_i \neq 1 $. Surprisingly, such phenomenon is still possible in this case, but quite rare. It occurs just on seven such groups of order less than $2\times 10^7$, namely, $PSp(4,3)$, $M_{12}$, $A_9$, $PSp(6,2)$, $M_{23}$, $PSU(5,2)$, $2F(4,2)'$. See the computation in Appendix.

We know that we can avoid $PSL(2,q)$, see here.

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSp{PSp}\DeclareMathOperator\PSU{PSU}$Background:

A representation $ \rho: G \to \GL(V) $ of a group $ G $ on a (complex) vector space $ V $ is said to be irreducible if there are no nontrivial invariant subspaces under the action of $ G $. A tensor product of two irreducible representations $ \rho_i: G \to \GL(V_i) $, $i=1,2$, is a representation on the tensor product space $ V_1 \otimes V_2 $, defined by $ \rho_1(g) \otimes \rho_2(g) $ for $ g \in G $.

To avoid such straightforward examples, let us assume that $G$ is a non-abelian finite simple group and $\rho_i \neq 1 $. Surprisingly, such phenomenon is still possible in this case, but quite rare. It occurs just on seven such groups of order less than $2\times 10^7$, namely, $\PSp(4,3)$, $M_{12}$, $A_9$, $\PSp(6,2)$, $M_{23}$, $\PSU(5,2)$, $2F(4,2)'$. See the computation in Appendix.

We know that we can avoid $\PSL(2,q)$, see here.