You could use pandas apply to obtain the intersection between the reference set and the set on each row of the dataframe. Then, apply the len function over the newly created column (Intersect) to quantify each intersection. This will enable you to filter the dataframe and only show rows where the set in sets intersects with another set (df['Len'] > 0).
df used as input
   id_one  id_two                                                                                         sets
0    7575     527          {1, 4, 6, 7, 8, 13, 16, 20, 24, 31, 40, 47, 50, 52, 57, 61, 64, 69, 80, 88, 91, 96}
1     574    1555      {7, 18, 19, 22, 23, 24, 30, 39, 43, 47, 50, 58, 62, 64, 72, 76, 77, 83, 84, 86, 87, 96}
2    7831    8823  {5, 14, 15, 20, 23, 28, 30, 32, 35, 36, 40, 41, 44, 52, 54, 59, 60, 62, 63, 84, 87, 90, 96}
3     688    6860           {2, 9, 20, 24, 27, 28, 30, 38, 46, 57, 59, 60, 64, 65, 69, 71, 80, 84, 88, 91, 95}
4    8843     596    {6, 7, 8, 24, 25, 27, 30, 33, 47, 50, 54, 56, 57, 61, 64, 66, 69, 74, 78, 81, 85, 88, 99}
5    1269    7546             {11, 22, 24, 25, 33, 35, 45, 48, 49, 54, 57, 59, 61, 68, 70, 75, 86, 87, 94, 95}
6    1362    4860           {2, 5, 14, 19, 23, 32, 37, 38, 47, 48, 58, 62, 65, 68, 70, 72, 73, 77, 82, 88, 91}
7    7994    7192      {2, 3, 4, 7, 9, 11, 12, 13, 15, 17, 20, 24, 25, 29, 40, 50, 57, 64, 71, 78, 89, 95, 99}
8     748    6271  {1, 12, 19, 26, 30, 34, 45, 48, 52, 60, 67, 72, 73, 74, 76, 80, 82, 84, 89, 94, 96, 98, 99}
9     553    4068  {9, 12, 15, 20, 35, 39, 40, 41, 44, 45, 50, 57, 65, 67, 68, 69, 72, 73, 79, 87, 88, 97, 98}
REFERENCE_SET = {10, 18, 23, 55, 90, 92}
df['Intersect'] = df['sets'].apply(lambda row: REFERENCE_SET.intersection(row))
df['Len'] = df['Intersect'].apply(len)
df_filtered = df[df['Len'] > 0]
Output from df_filtered
   id_one  id_two                                                                                         sets Intersect  Len
1     574    1555      {7, 18, 19, 22, 23, 24, 30, 39, 43, 47, 50, 58, 62, 64, 72, 76, 77, 83, 84, 86, 87, 96}  {18, 23}    2
2    7831    8823  {5, 14, 15, 20, 23, 28, 30, 32, 35, 36, 40, 41, 44, 52, 54, 59, 60, 62, 63, 84, 87, 90, 96}  {90, 23}    2
6    1362    4860           {2, 5, 14, 19, 23, 32, 37, 38, 47, 48, 58, 62, 65, 68, 70, 72, 73, 77, 82, 88, 91}      {23}    1