Assuming a three-column BED file you could do this, with an awk command as -wa -wb will make a 6-column BED file after the intersect:

% cat first.bed 
chr1    1   5
chr1    10  20
chr1    30  40

% cat second.bed 
chr1    4   8
chr1    15  18
chr1    100 200

% bedtools intersect -a first.bed -b second.bed -wa -wb                                                    
chr1    1   5   chr1    4   8
chr1    10  20  chr1    15  18

% bedtools intersect -a first.bed -b second.bed -wa -wb | awk 'OFS="\t" {print $1, $2, $3"\n"$4, $5, $6}'                                                 
chr1    1   5
chr1    4   8
chr1    10  20
chr1    15  18

# and finally all in one plus merge:
% bedtools intersect -a first.bed -b second.bed -wa -wb | awk 'OFS="\t" {print $1, $2, $3"\n"$4, $5, $6}' | sort -k1,1 -k2,2n | bedtools merge -i -
chr1    1   8
chr1    10  20

BEDtools version 2.30.0

Get all elements of first that have one or more overlaps with elements in second, and vice versa, and then merge those two sets:

% bedops --merge <(bedops -e 1 first.bed second.bed) <(bedops -e 1 second.bed first.bed)
chr1    1   8
chr1    10  20

Each of these is a process substitution that calculates overlaps:

<(bedops -e 1 first.bed second.bed)

and:

<(bedops -e 1 second.bed first.bed)

You can use process substitutions in place of files with BEDOPS tools.